Mathematical Methods in the Applied Sciences
Volume 48, Issue 12 pp. 12405-12420
RESEARCH ARTICLE
Open Access

A Stable Hybridized Discontinuous Galerkin Method for Solving Some Nonlinear m-Component Reaction–Diffusion Systems

Shima Baharlouei, 

Corresponding Author

Shima Baharlouei

Basque Center for Applied Mathematics (BCAM), Bilbao, Spain

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

Correspondence:

Shima Baharlouei ([email protected])

Contribution: Conceptualization, Writing - original draft, Writing - review & editing, ​Investigation, Visualization, Validation, Methodology, Software, Formal analysis

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Reza Mokhtari, 

Reza Mokhtari

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

Contribution: Writing - review & editing, ​Investigation, Supervision, Project administration

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Nabi Chegini, 

Nabi Chegini

Department of Mathematics, Tafresh University, Tafresh, Iran

Contribution: Writing - review & editing, Supervision

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First published: 12 May 2025
https://doi.org/10.1002/mma.11034

Funding: The authors received no specific funding for this work.

ABSTRACT

In this paper, we present a stable numerical scheme for solving two-dimensional m $$ m $$ -component reaction–diffusion systems. The proposed approach utilizes the backward Euler method for temporal discretization and the hybridized discontinuous Galerkin (HDG) method for spatial discretization. We analyze the stability of the proposed HDG method for problems with Dirichlet and Neumann boundary conditions, demonstrating that this method is stable, in the sense of the energy method, under certain mild conditions on the stabilization parameters. Several numerical experiments are provided to validate the proposed scheme, with applications to two reaction–diffusion models: the Brusselator and glycolysis systems. Numerical results confirm that the proposed method achieves the expected optimal convergence rate for both the approximate solutions and their first derivatives. To exhibit relevant physical concepts, we demonstrate the convergence behavior of approximate solutions at the stable equilibrium points of the selected reaction–diffusion system with small diffusion coefficients. Furthermore, the nonconvergence behavior is given at unstable equilibrium points.

1 Introduction

Investigating diverse scientific phenomena often relies on applying partial differential equations (PDEs). The inherent complexity of PDEs prompts a keen interest in the development of approximate solutions. Indeed, this pursuit arises from the recognition that, in many cases, analytical solutions may not be readily accessible. Also, calculating numerical approximations is a difficult task in terms of computational cost and convergence. This paper focuses on solving nonlinear reaction–diffusion systems, one of the most widely used and significant types of PDEs. This system finds extensive application across diverse scientific and engineering domains, with a particular emphasis on their utility in modeling chemical systems [1]. Moreover, it serves as a valuable tool for theoretical modeling in biological contexts. Their applications span predictions of morphological mechanisms in tissue and organ formation, modeling chemotaxis and cell polarity in cellular biology, and investigating properties related to cell polarity [2-4]. The initial form of the reaction–diffusion system has been extensively developed by Turing instability [2, 5-8], and despite their various dynamics, it provided suitable information about patterns. Therefore, it failed to model interfaces. On the other hand, the Allen–Cahn equation is a scalar reaction–diffusion model for investigating the interfaces in applied mathematics; see [9, 10]. In this paper, we address the following two-dimensional (2D) m $$ m $$ -component reaction–diffusion system [11]
∂ ∂ t u i = η i Δ u i + f i ( u 1 , … , u m ) , i = 1 , … , m , $$ \frac{\partial }{\partial t}{\mathfrak{u}}_i={\eta}_i\Delta {\mathfrak{u}}_i+{\mathfrak{f}}_i\left({\mathfrak{u}}_1,\dots, {\mathfrak{u}}_m\right),\kern2em i=1,\dots, m, $$ (1)
where u i = u i ( x , t ) , x = ( x 1 , x 2 ) , η i $$ {\mathfrak{u}}_i={\mathfrak{u}}_i\left(\mathbf{x},t\right),\kern0.3em \mathbf{x}=\left({x}_1,{x}_2\right),\kern0.3em {\eta}_i $$ is a positive constant, f i : ℝ m → ℝ $$ {\mathfrak{f}}_i:{\mathbb{R}}^m\to \mathbb{R} $$ is a given function, and Δ $$ \Delta $$ is the usual Laplacian operator over a rectangular-type domain Ω ⊂ ℝ 2 $$ \Omega \subset {\mathbb{R}}^2 $$ . To have a well-posed problem, system (1) is supplemented with appropriate initial and boundary conditions. Herein, the initial conditions are given by
u i ( x , 0 ) = u i , 0 , x ∈ Ω , i = 1 , … , m , $$ {\mathfrak{u}}_i\left(\mathbf{x},0\right)={\mathfrak{u}}_{i,0},\kern2em \mathbf{x}\in \Omega, \kern2em i=1,\dots, m, $$
and one of the following boundary conditions is imposed:
  • Dirichlet boundary conditions
    u i ( x , t ) = ϕ i , ( x , t ) ∈ ∂ Ω × ( 0 , T ] , i = 1 , 2 , … , m $$ {\mathfrak{u}}_i\left(\mathbf{x},t\right)={\phi}_i,\kern2em \left(\mathbf{x},t\right)\in \mathrm{\partial \Omega}\times \left(0,T\right],\kern2em i=1,2,\dots, m $$ (2)
  • Neumann boundary conditions
    ∂ ∂ n u i ( x , t ) = ψ i , ( x , t ) ∈ ∂ Ω × ( 0 , T ] , i = 1 , 2 , … , m $$ \frac{\partial }{\partial \mathbf{n}}{\mathfrak{u}}_i\left(\mathbf{x},t\right)={\psi}_i,\kern2em \left(\mathbf{x},t\right)\in \mathrm{\partial \Omega}\times \left(0,T\right],\kern2em i=1,2,\dots, m $$ (3)

where ϕ i = ϕ i ( x , t ) $$ {\phi}_i={\phi}_i\left(\mathbf{x},t\right) $$ and ψ i = ψ i ( x , t ) $$ {\psi}_i={\psi}_i\left(\mathbf{x},t\right) $$ are given functions and n $$ \mathbf{n} $$ is the outward unit normal vector. Assuming the positivity of the solution to the reaction–diffusion system (1) is grounded in its representation of essential physical concepts such as density and population sizes, which inherently possess positive attributes.

Although we cannot claim that the HDG method is the best DG method, as shown in [12, 13], we anticipate that the HDG method is an efficient numerical approach for solving (1). The first HDG method was introduced for solving second-order elliptic problems [14, 15]. Then, it was used significantly in solving many known and significant equations, such as the Ito-type coupled KdV system [16], coupled Burgers equations [17], the second-order elliptic equations with Dirac delta source [18], the convection–diffusion equation [19], the Navier–Stokes equations [20], some fractional PDEs [21-23], and many others [24, 25]. While it is possible to formulate a DG system with a reduced number of unknowns, the computational time required to solve these nonlinear systems is not as straightforward. The efficacy of the HDG method hinges on its distinctive approach to defining numerical fluxes, setting it apart from other DG methods, particularly the local discontinuous Galerkin (LDG) methods. In the HDG method, after reformulating the given system into the first-order system of equations along with its corresponding weak formulation, it is essential to define appropriate numerical fluxes. Numerical fluxes are defined in such a way that they introduce global unknowns, known as numerical traces, and combined with the local unknowns. Based on the types of numerical fluxes, a matrix–vector equation with a smaller bandwidth is generated, considering the first benefit of the HDG method. Also, imposing global unknowns in numerical traces reduces the degrees of freedom in the full-discretization system, which makes less computational time for solving the matrix–vector equation compared to other DG methods. Finally, to balance the number of equations and the unknowns, the continuous Galerkin (CG) method is exploited to define sufficient global equations. Regardless of the stated advantages of the HDG method, investigating stability is the main achievement of any DG method. For instance, system (1) with determined boundary conditions has been solved in [26] by using a specific kind of LDG method on the Cartesian grids. The stability of the proposed LDG method in [26] has been investigated by using the matrix method whereas, in this paper, the stability of the proposed HDG method is studied in the context of the energy method. We note that the proposed HDG method can be applied to all numerical examples in [26].

Various applicable systems across diverse fields of science and engineering can derive by selecting different forms of f i $$ {\mathfrak{f}}_i $$ in (1) for i = 1 , … , m $$ i=1,\dots, m $$ . Recently, some researchers have made efforts to solve these types of systems using various numerical methods [27-29]. This paper concentrates on validating the HDG method through numerical results, with a specific focus on applying it to the Brusselator and glycolysis systems. Using the modeling of autocatalytic chemical interactions [30], the Brusselator system has recently found extensive applications across various domains of chemistry and biology. Numerous numerical methods were employed to obtain the numerical solution of the Brusselator system. Notable approaches include the finite volume element method [30], the LDG method [31], and several others [32-34]. The glycolysis reaction–diffusion model is another type of the two-component reaction–diffusion system. The glycolysis model represents a biochemical reaction occurring within living cells [35, 36]. This paper presents a discussion of both the Brusselator and glycolysis systems.

The rest of the paper reads as follows. In Section 2, we introduce essential prerequisites, including approximation spaces, their associated inner products, and spatial partitioning. Section 3 is dedicated to the derivation of the HDG scheme. In Section 3, a fully discrete scheme corresponding to m $$ m $$ -component reaction–diffusion system (1) is obtained by using the backward Euler method and an HDG method for temporal and spatial discretizations, respectively. Also, the stability analysis of the proposed method is investigated. Additionally, Section 3 provides insights into the implementation, along with instructions regarding the linearization of the nonlinear system. Section 4 illustrates the effectiveness of the proposed HDG method by presenting some numerical experiments. Various tests in this section are conducted on two distinct reaction–diffusion systems: glycolysis and Brusselator systems. The optimal order of convergence for both systems is reported with respect to approximation spaces and generated meshes. Furthermore, this section demonstrates that the obtained approximate solutions exhibit convergence to the equilibrium points of the systems for small diffusion coefficients, especially as the time step size tends to zero. The final section is devoted to the conclusion.

2 Prerequisites

The first step of constructing an HDG method is to consider suitable meshes for the temporal and spatial domains. The time interval ( 0 , T ] $$ \left(0,T\right] $$ is partitioned into N $$ N $$ equal subintervals as
0 = t 0 < t 1 < ⋯ < t N = T $$ 0&#x0003D;{t}_0&lt;{t}_1&lt;\cdots &lt;{t}_N&#x0003D;T $$ (4)
where n = 0 , 1 , … , N , t n = n δ t $$ n&#x0003D;0,1,\dots, N,\kern0.3em {t}_n&#x0003D; n\delta t $$ where δ t = T / N $$ \delta t&#x0003D;T/N $$ is the time step size. The spatial domain Ω $$ \Omega $$ is partitioned into the set 𝒯 h = { K } as a collection of triangular elements K $$ K $$ where h $$ h $$ is the length of the longest edge among the edges of elements, and Ω = ⋃ K ∈ 𝒯 h K . This mesh generation is chosen in such a way that two arbitrary elements do not have any overlap. The set ∂ 𝒯 h = { ∂ K : K ∈ 𝒯 h } includes the boundaries of all elements where ∂ K $$ \partial K $$ denote the boundary of element K $$ K $$ . The set of all interior faces and boundary faces of ∂ Ω $$ \mathrm{\partial \Omega } $$ , respectively, are denoted by
E 0 : = { e : e = K ′ ∩ K , ∀ K , K ′ ∈ 𝒯 h , dim ( e ) = 1 , e ∉ ∂ Ω } ,
and E ∂ : = { e : e = K ∩ ∂ Ω , ∀ K ∈ 𝒯 h , dim ( e ) = 1 } . Also, the set E = E 0 ∪ E ∂ $$ E&#x0003D;{E}&#x0005E;0\cup {E}&#x0005E;{\partial } $$ is the collection of all faces. Note that, each member of E 0 $$ {E}&#x0005E;0 $$ is repeated and counted twice in ∂ 𝒯 h . The concept of jump for scalar-valued function v $$ v $$ on the interior faces e ∈ E 0 $$ e\in {E}&#x0005E;0 $$ is defined as
[ [ v ] ] = v + n + + v − n − , $$ \left[\left[v\right]\right]&#x0003D;{v}&#x0005E;{&#x0002B;}{\mathbf{n}}&#x0005E;{&#x0002B;}+{v}&#x0005E;{-}{\mathbf{n}}&#x0005E;{-}, $$
and on the boundary faces e ∈ E ∂ $$ e\in {E}&#x0005E;{\partial } $$ is defined as [ [ v ] ] = v n $$ \left[\left[v\right]\right]&#x0003D;v\mathbf{n} $$ . n + $$ {\mathbf{n}}&#x0005E;{&#x0002B;} $$ and n − $$ {\mathbf{n}}&#x0005E;{-} $$ denote outward unit normal vectors respect to the adjacent elements K + $$ {K}&#x0005E;{&#x0002B;} $$ and K − $$ {K}&#x0005E;{-} $$ with the common face e $$ e $$ ; see Figure 1.
Details are in the caption following the image
FIGURE 1
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Outward unit normal vectors n + $$ {\mathbf{n}}&#x0005E;{&#x0002B;} $$ and n − $$ {\mathbf{n}}&#x0005E;{-} $$ respect to the elements K + $$ {K}&#x0005E;{&#x0002B;} $$ and K − $$ {K}&#x0005E;{-} $$ . [Colour figure can be viewed at wileyonlinelibrary.com]
In the HDG method, two approximation spaces are needed, that is, for approximating the local unknowns on the elements and for approximating the global unknowns on the faces. Discontinuous finite element space for vector-valued functions is defined as
W h k , d = w ∈ ( L 2 ( Ω ) ) d : w | K ∈ ( 𝒫 k ( K ) ) d , ∀ K ∈ 𝒯 h ,
where d = 1 , 2 $$ d&#x0003D;1,2 $$ and 𝒫 k ( K ) is the set of all polynomial of degree at most k $$ k $$ over element K ∈ 𝒯 h . More precisely, W h k , d $$ {\mathbf{W}}_h&#x0005E;{k,d} $$ includes all piecewise polynomial functions over the generated mesh 𝒯 h . Skeleton space or trace space is defined as
M h k = { μ ∈ L 2 ( E ) : μ | e ∈ 𝒫 k ( e ) , ∀ e ∈ E } .
Also, applicable approximation subspace of M h k $$ {\mathbf{M}}_h&#x0005E;k $$ for performing the HDG method is defined as
M h k ( l ) = { μ ∈ M h k : μ ( x ) = Π l ( x ) , x ∈ ∂ Ω } , $$ {\mathbf{M}}_h&#x0005E;k(l)&#x0003D;\left\{\mu \in {\mathbf{M}}_h&#x0005E;k\kern0.3em :\kern0.3em \mu \left(\mathbf{x}\right)&#x0003D;\Pi \kern0.3em l\left(\mathbf{x}\right),\kern0.3em \mathbf{x}\in \mathrm{\partial \Omega}\right\}, $$
where l ∈ { ϕ i , ψ i } i = 1 m $$ l\in {\left\{{\phi}_i,{\psi}_i\right\}}_{i&#x0003D;1}&#x0005E;m $$ and Π $$ \Pi $$ is the L 2 $$ {L}&#x0005E;2 $$ projection into the skeleton space. In fact, the subspace M h k ( l ) $$ {M}_h&#x0005E;k(l) $$ is used to impose the projection of the given boundary data on ∂ Ω $$ \mathrm{\partial \Omega } $$ . Inner products of the spaces W h k , d $$ {\mathbf{W}}_h&#x0005E;{k,d} $$ and M h k $$ {\mathbf{M}}_h&#x0005E;k $$ are defined, respectively, as
( v , w ) 𝒯 h = ∫ 𝒯 h v · w d x = ∑ K ∈ 𝒯 h ∫ K v · w d x , ⟨ y , z ⟩ ∂ 𝒯 h = ∫ ∂ 𝒯 h y · z d s = ∑ ∂ K ∈ ∂ 𝒯 h ∫ ∂ K y · z d s ,
where “ · $$ \cdotp $$ ” is the inner product of two vector functions. Norms induced by the inner products are as follows:
‖ v ‖ L 2 ( 𝒯 h ) = ( v , v ) 𝒯 h , ‖ y ‖ L 2 ( ∂ 𝒯 h ) = ⟨ y , y ⟩ ∂ 𝒯 h .
In the following, the L 2 $$ {L}&#x0005E;2 $$ norms over 𝒯 h and ∂ 𝒯 h simply are denoted by ‖ · ‖ $$ \left\Vert \cdotp \right\Vert $$ .

3 Construction of the Numerical Scheme

This section is dedicated to the derivation of a fully discrete numerical scheme associated with the HDG approximation approach. Using the backward Euler method, a semidiscrete scheme can be achieved by discretization of the temporal variable. Then, a fully discrete form of system (1) is obtained by using the HDG method with defining suitable and relevant numerical fluxes/traces and stabilization parameters. Using the following backward finite difference scheme
∂ ∂ t u i ( x , t n ) ≃ u i n − u i n − 1 δ t $$ \frac{\partial }{\partial t}{\mathfrak{u}}_i\left(\mathbf{x},{t}_n\right)\simeq \frac{{\mathfrak{u}}_i&#x0005E;n-{\mathfrak{u}}_i&#x0005E;{n-1}}{\delta t} $$ (5)
where u i n ≃ u i ( x , t n ) $$ {\mathfrak{u}}_i&#x0005E;n\simeq {\mathfrak{u}}_i\left(\mathbf{x},{t}_n\right) $$ , we get the following semidiscrete scheme corresponding to (1)
u i n − δ t η i Δ u i n − δ t f i n = u i n − 1 , i = 1 , … , m $$ {\mathfrak{u}}_i&#x0005E;n-\delta t\kern0.3em {\eta}_i\kern0.3em \Delta {\mathfrak{u}}_i&#x0005E;n-\delta t\kern0.3em {\mathfrak{f}}_i&#x0005E;n&#x0003D;{\mathfrak{u}}_i&#x0005E;{n-1},\kern2em i&#x0003D;1,\dots, m $$ (6)
The next step is to find the corresponding first-order system of equations with respect to time partitioning. Consider the auxiliary variables q i 1 n = η i ∂ ∂ x 1 u i n $$ {\mathfrak{q}}_{i1}&#x0005E;n&#x0003D;{\eta}_i\frac{\partial }{\partial {x}_1}{\mathfrak{u}}_i&#x0005E;n $$ and q i 2 n = η i ∂ ∂ x 2 u i n $$ {\mathfrak{q}}_{i2}&#x0005E;n&#x0003D;{\eta}_i\frac{\partial }{\partial {x}_2}{\mathfrak{u}}_i&#x0005E;n $$ . A first-order system of equations corresponding to (6) is as follows:
u i n − δ t ∇ · q i n − δ t f i n = u i n − 1 , q i n − ∇ u i n = 0 , $$ {\displaystyle \begin{array}{cc}\hfill {\mathfrak{u}}_i&#x0005E;n-\delta t\kern0.3em \nabla \cdotp {\mathfrak{q}}_i&#x0005E;n-\delta t\kern0.3em {\mathfrak{f}}_i&#x0005E;n&#x0003D;&amp; {\mathfrak{u}}_i&#x0005E;{n-1},\hfill \\ {}\hfill {\mathfrak{q}}_i&#x0005E;n-\nabla {\mathfrak{u}}_i&#x0005E;n&#x0003D;&amp; 0,\hfill \end{array}} $$ (7)
where q i n = ( q i 1 n , q i 2 n ) T $$ {\mathfrak{q}}_i&#x0005E;n&#x0003D;{\left({\mathfrak{q}}_{i1}&#x0005E;n,{\mathfrak{q}}_{i2}&#x0005E;n\right)}&#x0005E;T $$ . Let u i n , q i 1 n , q i 2 n , q i n = ( q i 1 n , q i 2 n ) T $$ {u}_i&#x0005E;n,\kern0.3em {q}_{i1}&#x0005E;n,\kern0.3em {q}_{i2}&#x0005E;n,\kern0.3em {q}_i&#x0005E;n&#x0003D;{\left({q}_{i1}&#x0005E;n,{q}_{i2}&#x0005E;n\right)}&#x0005E;T $$ , and f i n = f i ( u 1 n , … , u m n ) $$ {f}_i&#x0005E;n&#x0003D;{\mathfrak{f}}_i\left({u}_1&#x0005E;n,\dots, {u}_m&#x0005E;n\right) $$ be numerical approximations of u i n , q i 1 n , q i 2 n , q i n $$ {\mathfrak{u}}_i&#x0005E;n,\kern0.3em {\mathfrak{q}}_{i1}&#x0005E;n,\kern0.3em {\mathfrak{q}}_{i2}&#x0005E;n,\kern0.3em {\mathfrak{q}}_i&#x0005E;n $$ , and f i n $$ {\mathfrak{f}}_i&#x0005E;n $$ , respectively. In the corresponding weak formulation of (7), we seek numerical approximation
( u 1 n , … , u m n , q 1 n , … , q m n ) ∈ × W h k , 1 × ⋯ × W h k , 1 × W h k , 2 × ⋯ × W h k , 2 , $$ \left({u}_1&#x0005E;n,\dots, {u}_m&#x0005E;n,{q}_1&#x0005E;n,\dots, {q}_m&#x0005E;n\right)\in \times {\mathbf{W}}_h&#x0005E;{k,1}\times \cdots \times {\mathbf{W}}_h&#x0005E;{k,1}\times {\mathbf{W}}_h&#x0005E;{k,2}\times \cdots \times {\mathbf{W}}_h&#x0005E;{k,2}, $$
such that for all test functions
( w 1 , … , w m , v 1 , … , v m ) ∈ × W h k , 1 × ⋯ × W h k , 1 × W h k , 2 × ⋯ × W h k , 2 , $$ \left({w}_1,\dots, {w}_m,{v}_1,\dots, {v}_m\right)\in \times {\mathbf{W}}_h&#x0005E;{k,1}\times \cdots \times {\mathbf{W}}_h&#x0005E;{k,1}\times {\mathbf{W}}_h&#x0005E;{k,2}\times \cdots \times {\mathbf{W}}_h&#x0005E;{k,2}, $$
and all K ∈ 𝒯 h , it holds that
( u i n , w i ) K + δ t ( q i n , ∇ w i ) K − δ t ( f i n , w i ) K − δ t ⟨ q i n ^ n , w i ⟩ ∂ K = ( u i n − 1 , w i ) K , ( q i n , v i ) K + ( u i n , ∇ · v i ) K − ⟨ û i n n , v i ⟩ ∂ K = 0 , $$ {\displaystyle \begin{array}{cc}\hfill {\left({u}_i&#x0005E;n,{w}_i\right)}_K&amp; &#x0002B;\delta t{\left({q}_i&#x0005E;n,\nabla {w}_i\right)}_K-\delta t{\left({f}_i&#x0005E;n,{w}_i\right)}_K\hfill \\ {}\hfill &amp; -\delta t{\left\langle \hat{q_i&#x0005E;n}\mathbf{n},{w}_i\right\rangle}_{\partial K}&#x0003D;{\left({u}_i&#x0005E;{n-1},{w}_i\right)}_K,\hfill \\ {}\hfill {\left({q}_i&#x0005E;n,{v}_i\right)}_K&amp; &#x0002B;{\left({u}_i&#x0005E;n,\nabla \cdotp {v}_i\right)}_K-{\left\langle {\hat{u}}_i&#x0005E;n\mathbf{n},{v}_i\right\rangle}_{\partial K}&#x0003D;0,\hfill \end{array}} $$ (8)
where q i n ^ $$ \hat{q_i&#x0005E;n} $$ and û i n $$ {\hat{u}}_i&#x0005E;n $$ are numerical flux and numerical trace, respectively. The numerical flux q i n ^ $$ \hat{q_i&#x0005E;n} $$ is defined as
q i n ^ = q i n + τ i ( u i n − û i n ) n , i = 1 , … , m , $$ \hat{q_i&#x0005E;n}&#x0003D;{q}_i&#x0005E;n&#x0002B;{\tau}_i\left({u}_i&#x0005E;n-{\hat{u}}_i&#x0005E;n\right)\mathbf{n},\kern2em i&#x0003D;1,\dots, m, $$ (9)
where the stabilization parameter τ i $$ {\tau}_i $$ is responsible for the stability of the numerical scheme. Although definition of the numerical fluxes is not unique, their appropriate definition is needed to guarantee stability of the method. In the HDG scheme, boundary conditions are imposed to the definition of the numerical traces. So the numerical trace û i ∈ M h k ( ϕ i ) $$ {\hat{u}}_i\in {\mathbf{M}}_h&#x0005E;k\left({\phi}_i\right) $$ at time t n $$ {t}_n $$ is defined as
û i n = ϕ i n , e ∈ E ∂ , λ i n , e ∈ E 0 , $$ {\hat{u}}_i&#x0005E;n&#x0003D;\left\{\begin{array}{ll}{\phi}_i&#x0005E;n,&amp; e\in {E}&#x0005E;{\partial },\\ {}{\lambda}_i&#x0005E;n,&amp; e\in {E}&#x0005E;0,\end{array}\right. $$ (10)
where ϕ i n $$ {\phi}_i&#x0005E;n $$ is Dirichlet boundary data (2) and λ i n ∈ M h k ( 0 ) $$ {\lambda}_i&#x0005E;n\in {\mathbf{M}}_h&#x0005E;k(0) $$ for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ are global unknowns at time level n $$ n $$ which are added to the local unknowns u i n $$ {u}_i&#x0005E;n $$ and q i n $$ {q}_i&#x0005E;n $$ . We note that the boundary conditions are imposed on the boundary faces and numerical trace û i n $$ {\hat{u}}_i&#x0005E;n $$ acts on the interior faces as the global unknown λ i n $$ {\lambda}_i&#x0005E;n $$ . Thus, it is needed to have m $$ m $$ extra global equations for adding to weak formulation (8), which is obtained by applying conservation of the numerical fluxes. Indeed, we achieve the following global equations by imposing the zero jump to the numerical fluxes (9) on all interior faces,
[ [ q i n ^ · n ] ] = 0 , for e ∈ E 0 $$ \left[\left[\hat{q_i&#x0005E;n}\cdotp \mathbf{n}\right]\right]&#x0003D;0,\kern2em \mathrm{for}\kern0.60em e\in {E}&#x0005E;0 $$ (11)
Summing Equations (8) and (11) over all elements and using (9) and (10), we get the following system of equations:
( u i n , w i ) 𝒯 h − δ t ( ∇ · q i n , w i ) 𝒯 h − δ t ( f i n , w i ) 𝒯 h − δ t ⟨ τ i u i n , w i ⟩ ∂ 𝒯 h + δ t ⟨ τ i λ i n , w i ⟩ ∂ 𝒯 h ∖ E ∂ = l 1 ( w i ) , ( q i n , v i ) 𝒯 h + ( u i n , ∇ · v i ) 𝒯 h − ⟨ λ i n n , v i ⟩ ∂ 𝒯 h ∖ E ∂ = l 2 ( v i ) , ⟨ q i n · n , μ i ⟩ ∂ 𝒯 h ∖ E ∂ + ⟨ τ i u i n , μ i ⟩ ∂ 𝒯 h ∖ E ∂ − ⟨ τ i λ i n , μ i ⟩ ∂ 𝒯 h ∖ E ∂ = 0 , (12)
where ( μ 1 , … , μ m ) ∈ M h k ( 0 ) × ⋯ × M h k ( 0 ) $$ \left({\mu}_1,\dots, {\mu}_m\right)\in {M}_h&#x0005E;k(0)\times \cdots \times {M}_h&#x0005E;k(0) $$ and
l 1 ( w i ) = ( u i n − 1 , w i ) 𝒯 h − δ t ⟨ τ i ϕ i n , w i ⟩ E ∂ , l 2 ( v i ) = ⟨ ϕ i n n , v i ⟩ E ∂ .

Remark 1.If system (1) is equipped with the Neumann boundary conditions (3), because there is not any boundary value for u i $$ {\mathfrak{u}}_i $$ , then definition of the numerical traces (10) changes to

û i n = λ i n , for e ∈ E , i = 1 , … , m $$ {\hat{u}}_i&#x0005E;n&#x0003D;{\lambda}_i&#x0005E;n,\kern1em \mathrm{for}\kern0.60em e\in E,\kern2em i&#x0003D;1,\dots, m $$ (13)

According to (13), it is needed to have m $$ m $$ global equations which act on all faces with imposing the Neumann boundary conditions, that is,

q i n · n = ψ i n , e ∈ E ∂ , [ [ q i n ^ · n ] ] = 0 , e ∈ E 0 . $$ \left\{\begin{array}{ll}{q}_i&#x0005E;n\cdotp \mathbf{n}&#x0003D;{\psi}_i&#x0005E;n,&amp; e\in {E}&#x0005E;{\partial },\\ {}\left[\left[\hat{q_i&#x0005E;n}\cdotp \mathbf{n}\right]\right]&#x0003D;0,&amp; e\in {E}&#x0005E;0.\end{array}\right. $$ (14)

With summing Equations (8) and (14) over all elements and using (9) and (13), we get the following system for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ ,

( u i n , w i ) 𝒯 h − δ t ( ∇ · q i n , w i ) 𝒯 h − δ t ( f i n , w i ) 𝒯 h − δ t ⟨ τ i u i n , w i ⟩ ∂ 𝒯 h + δ t ⟨ τ i λ i n , w i ⟩ ∂ 𝒯 h = l 1 ( w i ) , ( q i n , v i ) 𝒯 h + ( u i n , ∇ · v i ) 𝒯 h − ⟨ λ i n n , v i ⟩ ∂ 𝒯 h = 0 , ⟨ q i n · n , μ i ⟩ ∂ 𝒯 h + ⟨ τ i u i n , μ i ⟩ ∂ 𝒯 h − ⟨ τ i λ i n , μ i ⟩ ∂ 𝒯 h = l 2 ( μ i ) ,
where ( μ 1 , … , μ m ) ∈ M h k × ⋯ × M h k $$ \left({\mu}_1,\dots, {\mu}_m\right)\in {M}_h&#x0005E;k\times \cdots \times {M}_h&#x0005E;k $$ and
l 1 ( w i ) = ( u i n − 1 , w i ) 𝒯 h , l 2 ( μ i ) = − ⟨ ψ i n n , μ i ⟩ E ∂ .

3.1 Stability Analysis

In this section, we focus on investigation of stability of the proposed HDG method. In studying reaction–diffusion system (1), some specific assumptions have to be considered over f i $$ {\mathfrak{f}}_i $$ . Without loss of generality, we assume that f i $$ {\mathfrak{f}}_i $$ is bounded over the spatial domain Ω $$ \Omega $$ , that is, there are positive constants L 1 , … , L m $$ {L}_1,\dots, {L}_m $$ such that
‖ f i ( v ) ‖ ≤ L i ‖ v ‖ , i = 1 , … , m , v = [ v 1 , … , v m ] T . $$ \left\Vert {\mathfrak{f}}_i(v)\right\Vert \le {L}_i\left\Vert v\right\Vert, \kern2em i&#x0003D;1,\dots, m,\kern2em v&#x0003D;{\left[{v}_1,\dots, {v}_m\right]}&#x0005E;T. $$

Theorem 1.If m $$ m $$ -component reaction–diffusion system (1) is equipped with the homogeneous Dirichlet boundary conditions, then the proposed HDG method is stable provided that τ i < 0 , i = 1 , … , m $$ {\tau}_i&lt;0,\kern0.3em i&#x0003D;1,\dots, m $$ .

Proof.By setting w i = u i n , v i = δ t q i n $$ {w}_i&#x0003D;{u}_i&#x0005E;n,\kern0.3em {v}_i&#x0003D;\delta t\kern0.3em {q}_i&#x0005E;n $$ in (8), for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ , we have

( u i n , u i n ) K + δ t ( q i n , ∇ u i n ) K − δ t ( f i n , u i n ) K − δ t ⟨ q i n ^ n , u i n ⟩ ∂ K = ( u i n − 1 , u i n ) K , δ t ( q i n , q i n ) K + δ t ( u i n , ∇ · q i n ) K − δ t ⟨ û i n n , q i n ⟩ ∂ K = 0 . $$ {\displaystyle \begin{array}{cc}&amp; \hfill {\left({u}_i&#x0005E;n,{u}_i&#x0005E;n\right)}_K&#x0002B;\delta t{\left({q}_i&#x0005E;n,\nabla {u}_i&#x0005E;n\right)}_K-\delta t{\left({f}_i&#x0005E;n,{u}_i&#x0005E;n\right)}_K\\ {}&amp; -\delta t{\left\langle \hat{q_i&#x0005E;n}\mathbf{n},{u}_i&#x0005E;n\right\rangle}_{\partial K}&#x0003D;{\left({u}_i&#x0005E;{n-1},{u}_i&#x0005E;n\right)}_K,\hfill \\ {}\hfill \delta t{\left({q}_i&#x0005E;n,{q}_i&#x0005E;n\right)}_K&#x0002B;\delta t{\left({u}_i&#x0005E;n,\nabla \cdotp {q}_i&#x0005E;n\right)}_K-\delta t{\left\langle {\hat{u}}_i&#x0005E;n\mathbf{n},{q}_i&#x0005E;n\right\rangle}_{\partial K}&#x0003D;&amp; \hfill 0.\end{array}} $$ (15)

By summing Equation (15), we get

∑ i = 1 m ‖ u i n ‖ L 2 ( K ) 2 + δ t ∑ i = 1 m ‖ q i n ‖ L 2 ( K ) 2 + δ t ∑ i = 1 m Θ i K = δ t ∑ i = 1 m ( f i n , u i n ) K + ∑ i = 1 m ( u i n − 1 , u i n ) K , $$ {\displaystyle \begin{array}{cc}\hfill \sum \limits_{i&#x0003D;1}&#x0005E;m{\left\Vert {u}_i&#x0005E;n\right\Vert}_{L&#x0005E;2(K)}&#x0005E;2&amp; &#x0002B;\delta t\sum \limits_{i&#x0003D;1}&#x0005E;m{\left\Vert {q}_i&#x0005E;n\right\Vert}_{L&#x0005E;2(K)}&#x0005E;2&#x0002B;\delta t\sum \limits_{i&#x0003D;1}&#x0005E;m{\Theta}_i&#x0005E;K&#x0003D;\delta t\sum \limits_{i&#x0003D;1}&#x0005E;m{\left({f}_i&#x0005E;n,{u}_i&#x0005E;n\right)}_K\hfill \\ {}\hfill &amp; &#x0002B;\sum \limits_{i&#x0003D;1}&#x0005E;m{\left({u}_i&#x0005E;{n-1},{u}_i&#x0005E;n\right)}_K,\hfill \end{array}} $$
or equivalently
‖ u n ‖ L 2 ( K ) 2 + δ t ‖ q n ‖ L 2 ( K ) 2 + δ t ∑ i = 1 m Θ i K = δ t ( f n , u n ) K + ( u n − 1 , u n ) K $$ {\left\Vert {u}&#x0005E;n\right\Vert}_{L&#x0005E;2(K)}&#x0005E;2&#x0002B;\delta t{\left\Vert {q}&#x0005E;n\right\Vert}_{L&#x0005E;2(K)}&#x0005E;2&#x0002B;\delta t\sum \limits_{i&#x0003D;1}&#x0005E;m{\Theta}_i&#x0005E;K&#x0003D;\delta t{\left({f}&#x0005E;n,{u}&#x0005E;n\right)}_K&#x0002B;{\left({u}&#x0005E;{n-1},{u}&#x0005E;n\right)}_K $$ (16)
where u n = ( u 1 n , … , u m n ) , q n = ( q 1 n , … , q m n ) , f n = ( f 1 n , … , f m n ) $$ {u}&#x0005E;n&#x0003D;\left({u}_1&#x0005E;n,\dots, {u}_m&#x0005E;n\right),\kern0.3em {q}&#x0005E;n&#x0003D;\left({q}_1&#x0005E;n,\dots, {q}_m&#x0005E;n\right),\kern0.3em {f}&#x0005E;n&#x0003D;\left({f}_1&#x0005E;n,\dots, {f}_m&#x0005E;n\right) $$ , and
Θ i K = ( q i n , ∇ u i n ) K + ( u i n , ∇ · q i n ) K − ⟨ q i n ^ n , u i n ⟩ ∂ K − ⟨ û i n n , q i n ⟩ ∂ K . $$ {\Theta}_i&#x0005E;K&#x0003D;{\left({q}_i&#x0005E;n,\nabla {u}_i&#x0005E;n\right)}_K&#x0002B;{\left({u}_i&#x0005E;n,\nabla \cdotp {q}_i&#x0005E;n\right)}_K-{\left\langle \hat{q_i&#x0005E;n}\mathbf{n},{u}_i&#x0005E;n\right\rangle}_{\partial K}-{\left\langle {\hat{u}}_i&#x0005E;n\mathbf{n},{q}_i&#x0005E;n\right\rangle}_{\partial K}. $$ (17)

Using the divergence theorem, we have

( q i n , ∇ u i n ) K + ( u i n , ∇ · q i n ) K = ∫ K ∇ · ( q i n u i n ) d x = ∫ ∂ K ( q i n u i n ) · n d s = ⟨ q i n · n , u i n ⟩ ∂ K . $$ {\displaystyle \begin{array}{cc}&amp; \hfill {\left({q}_i&#x0005E;n,\nabla {u}_i&#x0005E;n\right)}_K&#x0002B;{\left({u}_i&#x0005E;n,\nabla \cdotp {q}_i&#x0005E;n\right)}_K&#x0003D;{\int}_K\nabla \cdotp \left({q}_i&#x0005E;n{u}_i&#x0005E;n\right)\kern0.3em d\mathbf{x}\\ {}\hfill &amp; &#x0003D;{\int}_{\partial K}\left({q}_i&#x0005E;n{u}_i&#x0005E;n\right)\cdotp \mathbf{n}\kern0.3em ds&#x0003D;{\left\langle {q}_i&#x0005E;n\cdotp \mathbf{n},{u}_i&#x0005E;n\right\rangle}_{\partial K}.\hfill \end{array}} $$ (18)

Applying (18) to (17) and using

⟨ û i n n , q i n ⟩ ∂ K = ⟨ q i n · n , û i n ⟩ ∂ K $$ {\left\langle {\hat{u}}_i&#x0005E;n\mathbf{n},{q}_i&#x0005E;n\right\rangle}_{\partial K}&#x0003D;{\left\langle {q}_i&#x0005E;n\cdotp \mathbf{n},{\hat{u}}_i&#x0005E;n\right\rangle}_{\partial K} $$
lead to
Θ i K = − ⟨ q i n ^ · n , u i n ⟩ ∂ K + ⟨ q i n · n , u i n − û i n ⟩ ∂ K . $$ {\Theta}_i&#x0005E;K&#x0003D;-{\left\langle \hat{q_i&#x0005E;n}\cdotp \mathbf{n},{u}_i&#x0005E;n\right\rangle}_{\partial K}&#x0002B;{\left\langle {q}_i&#x0005E;n\cdotp \mathbf{n},{u}_i&#x0005E;n-{\hat{u}}_i&#x0005E;n\right\rangle}_{\partial K}. $$ (19)

Because û i n $$ {\hat{u}}_i&#x0005E;n $$ and q i n ^ $$ \hat{q_i&#x0005E;n} $$ are single-value functions and ϕ i = 0 $$ {\phi}_i&#x0003D;0 $$ , for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ , we conclude that

∑ K ∈ 𝒯 h ⟨ q i n ^ · n , û i n ⟩ ∂ K = 0 . (20)

By substituting (20) in the sum of (19) over all elements, we get

∑ K ∈ 𝒯 h Θ i K = ∑ K ∈ 𝒯 h ( − ⟨ q i n ^ · n , u i n − û i n ⟩ ∂ K + ⟨ q i n · n , u i n − û i n ⟩ ∂ K ) = ∑ K ∈ 𝒯 h ⟨ ( q i n − q i n ^ ) · n , u i n − û i n ⟩ ∂ K . (21)

Using the definition of numerical flux q i n ^ $$ \hat{q_i&#x0005E;n} $$ from (9) in (21), we obtain

∑ K ∈ 𝒯 h Θ i K = ∑ K ∈ 𝒯 h ⟨ − τ i , ( u i n − û i n ) 2 ⟩ ∂ K = ⟨ − τ i , ( u i n − û i n ) 2 ⟩ ∂ 𝒯 h .
According to the assumption of the theorem, that is, τ i < 0 $$ {\tau}_i&lt;0 $$ , for i = 1 , ⋯ , m $$ i&#x0003D;1,\cdots \kern0.3em ,m $$ , it can be concluded that
∑ i = 1 m ∑ K ∈ 𝒯 h Θ i K > 0 (22)

Summing (16) over all elements, and then using ∑ K ∈ 𝒯 h ‖ q n ‖ L 2 ( K ) ≥ 0 and (22), we get

∑ K ∈ 𝒯 h ‖ u n ‖ L 2 ( K ) 2 ≤ δ t ∑ K ∈ 𝒯 h ( f n , u n ) K + ∑ K ∈ 𝒯 h ( u n , u n − 1 ) K ,
or equivalently
‖ u n ‖ 2 ≤ δ t ( f n , u n ) 𝒯 h + ( u n , u n − 1 ) 𝒯 h . (23)

Using the Cauchy–Schwarz inequality, we have

‖ u n ‖ ≤ δ t ‖ f n ‖ + ‖ u n − 1 ‖ $$ \left\Vert {u}&#x0005E;n\right\Vert \le \delta t\left\Vert {f}&#x0005E;n\right\Vert &#x0002B;\left\Vert {u}&#x0005E;{n-1}\right\Vert $$ (24)
which by assuming δ t $$ \delta t $$ to be sufficiently small, and by using the boundedness of f i $$ {\mathfrak{f}}_i $$ 's, it leads to
‖ u n ‖ ≤ C ‖ u 0 ‖ . $$ \left\Vert {u}&#x0005E;n\right\Vert \le C\left\Vert {u}&#x0005E;0\right\Vert . $$

Corollary 1.If m $$ m $$ -component reaction–diffusion system (1) is equipped with the homogeneous Neumann boundary conditions then the proposed HDG method is stable provided that τ i < 0 , i = 1 , … , m $$ {\tau}_i&lt;0,\kern0.3em i&#x0003D;1,\dots, m $$ .

Proof.Based on the definition of global Equation (14) and ψ i n = 0 $$ {\psi}_i&#x0005E;n&#x0003D;0 $$ , we get

∑ K ∈ 𝒯 h ⟨ q i n ^ · n , û i n ⟩ ∂ K = 0 .
The rest of the proof is similar to Theorem 1.

3.2 Linearization Technique

The fully discrete form of (12) has been presented based on the backward Euler method and HDG method, respectively, for the time and space discretization. The aim of this section is to solve the nonlinear system (12). So it is needed to linearize the nonlinear terms or exploit an appropriate numerical method for solving nonlinear systems, for instance, Newton–Raphson method. Using the Taylor series expansion to each f i n $$ {f}_i&#x0005E;n $$ , for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ , leads to the following second-order linearization formula:
f i n = f i ( u 1 n , … , u m n ) = f i n − 1 + ( u 1 n − u 1 n − 1 ) ∂ ∂ u 1 f i n − 1 + ⋯ + ( u m n − u m n − 1 ) ∂ ∂ u m f i n − 1 + O ( δ t 2 ) . $$ {\displaystyle \begin{array}{cc}\hfill {f}_i&#x0005E;n&amp; &#x0003D;{\mathfrak{f}}_i\left({u}_1&#x0005E;n,\dots, {u}_m&#x0005E;n\right)&#x0003D;{f}_i&#x0005E;{n-1}&#x0002B;\left({u}_1&#x0005E;n-{u}_1&#x0005E;{n-1}\right)\frac{\partial }{\partial {u}_1}{f}_i&#x0005E;{n-1}\hfill \\ {}\hfill &amp; \kern10pt &#x0002B;\cdots &#x0002B;\left({u}_m&#x0005E;n-{u}_m&#x0005E;{n-1}\right)\frac{\partial }{\partial {u}_m}{f}_i&#x0005E;{n-1}&#x0002B;O\left(\delta {t}&#x0005E;2\right).\hfill \end{array}} $$ (25)
Although it is enough to use a first-order linearization technique, we apply a second-order technique just to be sure that this linearization does not affect the temporal order of accuracy of the method. Using (25) into (12), for i = 1 , … , m $$ i&#x0003D;1,\dots, m $$ , we have
a i ( u i n , w i ) − δ t ∑ j = 1 m b i j ( u j n , w i ) − δ t c i ( u i n , w i ) − δ t d i ( q i n , w i ) + δ t e i ( λ i n , w i ) = l 1 ( w i ) , a i ( q i n , v i ) + d i T ( u i n , v i ) 𝒯 h − f i ( λ i n , v i ) = l 2 ( v i ) , g i ( u i n , μ i ) + h i ( q i n , μ i ) − k i ( λ i n , μ i ) = 0 , (26)
where
a i ( u i n , w ) = ( u i n , w ) 𝒯 h , b i j ( u j n , w ) = ( u j n ∂ ∂ u j f i n − 1 , w ) 𝒯 h , c i ( u i n , w ) = ⟨ τ i u i n , w ⟩ ∂ 𝒯 h , d i ( q i n , w ) = ( ∇ · q i n , w ) 𝒯 h , e i ( λ i n , w ) = ⟨ τ i λ i n , w ⟩ ∂ 𝒯 h ∖ E ∂ , f i ( λ i n , w ) = ⟨ λ i n n , w ⟩ ∂ 𝒯 h ∖ E ∂ , g i ( u i n , μ ) = ⟨ τ i u i n , μ ⟩ ∂ 𝒯 h ∖ E ∂ , h i ( q i n , μ ) = ⟨ q i n · n , μ ⟩ ∂ 𝒯 h ∖ E ∂ , k i ( λ i n , μ ) = ⟨ τ i λ i n , μ ⟩ ∂ 𝒯 h ∖ E ∂ , l 2 ( w ) = ⟨ ϕ i n n , w ⟩ E ∂ l 1 ( w ) = δ t − f i n − 1 + u 1 n − 1 ∂ ∂ u 1 f i n − 1 + ⋯ + u m n − 1 ∂ ∂ u m f i n − 1 , w 𝒯 h + ( u i n − 1 , w ) 𝒯 h − δ t ⟨ τ i ϕ i n , w ⟩ E ∂ .
The corresponding matrix–vector equation of (26) has not to be solved globally because of its high computational costs. Therefore, it is suggested to use the Schur complement technique that is raised in solving a system of linear equations. At first, the matrix–vector equation of (26) should be rewritten in the form of a 4 × 4 $$ 4\times 4 $$ block matrix, and then, Schur complement technique leads to solve two smaller systems instead of solving a large system (26); see [16]. The idea of applying the Schur complement is not novel, and it is actually the basic HDG static condensation/local elimination but extended towards a system of equations. This utilized technique has been used successfully in previous works [37-39] and some more performance-oriented works [12, 40]. Substructuring techniques used in [40] can speed up the inversion of the block matrices which is relevant for variable-coefficient equations and can be applied to nonlinear system (1).

4 Numerical Simulations

In this section, we aim to evaluate the performance of the proposed HDG method in solving various systems of nonlinear reaction–diffusion equations, specifically the Brusselator and glycolysis systems. In Section 4.1, we investigate the Brusselator system, analyzing its properties and presenting numerical results through three examples. In the first example, the HDG method demonstrates the optimal order of convergence for both the approximate solutions and their first derivatives with respect to the spatial variables. Example 2 examines cases where analytical solutions are unavailable, showcasing the method's convergence towards equilibrium points. Example 3 further investigates two Brusselator systems, highlighting stable and unstable equilibrium points. Section 4.2 introduces the glycolysis system, followed by a performance assessment of the proposed HDG method in three examples. First, we confirm the optimal order of convergence for both the approximate solutions and their first derivatives. Then, in the fifth example, we explore the convergence of numerical results towards equilibrium points for systems with varying stability. Finally, we use the proposed HDG method to solve a three-component system. Note that, in all tests, we employ triangular meshes generated using the Delaunay triangulation algorithm.

4.1 Brusselator System

Brusselator system, a two-component reaction–diffusion system, is derived by considering system (1) with m = 2 , f 1 ( u 1 , u 2 ) = u 1 2 u 2 − ( ξ + 1 ) u 1 + γ , f 2 ( u 1 , u 2 ) = − u 1 2 u 2 + ξ u 1 $$ m&#x0003D;2,\kern0.3em {\mathfrak{f}}_1\left({\mathfrak{u}}_1,{\mathfrak{u}}_2\right)&#x0003D;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2-\left(\xi &#x0002B;1\right){\mathfrak{u}}_1&#x0002B;\gamma, \kern0.3em {\mathfrak{f}}_2\left({\mathfrak{u}}_1,{\mathfrak{u}}_2\right)&#x0003D;-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;\xi {\mathfrak{u}}_1 $$ , and η 1 = η 2 = η $$ {\eta}_1&#x0003D;{\eta}_2&#x0003D;\eta $$ . See [32, 33]. In this form, the Brusselator system is expressed as
∂ ∂ t u 1 = η Δ u 1 + u 1 2 u 2 − ( ξ + 1 ) u 1 + γ , ∂ ∂ t u 2 = η Δ u 2 − u 1 2 u 2 + ξ u 1 , $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1&amp; &#x0003D;\eta \Delta {\mathfrak{u}}_1&#x0002B;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2-\left(\xi &#x0002B;1\right){\mathfrak{u}}_1&#x0002B;\gamma, \hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2&amp; &#x0003D;\eta \Delta {\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;\xi {\mathfrak{u}}_1,\hfill \end{array}} $$ (27)
where ξ $$ \xi $$ and γ $$ \gamma $$ are real value constants, and x = ( x 1 , x 2 ) ∈ Ω ⊂ ℝ 2 $$ \mathbf{x}&#x0003D;\left({x}_1,{x}_2\right)\in \Omega \subset {\mathbb{R}}&#x0005E;2 $$ . For the small value of η $$ \eta $$ , the dissipative terms in system (27) become negligible, which leads to the following system of ordinary differential equations:
∂ ∂ t u 1 ≃ u 1 2 u 2 − ( ξ + 1 ) u 1 + γ , ∂ ∂ t u 2 ≃ − u 1 2 u 2 + ξ u 1 . $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1&amp; \simeq {\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2-\left(\xi &#x0002B;1\right){\mathfrak{u}}_1&#x0002B;\gamma, \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2&amp; \simeq -{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;\xi {\mathfrak{u}}_1.\end{array}} $$ (28)

System (28) has a single equilibrium (critical) point at ( γ , ξ / γ ) $$ \left(\gamma, \xi /\gamma \right) $$ . If γ $$ \gamma $$ and ξ $$ \xi $$ are chosen such that 1 − ξ + γ 2 > 0 $$ 1-\xi &#x0002B;{\gamma}&#x0005E;2&gt;0 $$ , it can be shown that the corresponding equilibrium point is stable, and the steady-state solutions ( u 1 , u 2 ) $$ \left({\mathfrak{u}}_1,{\mathfrak{u}}_2\right) $$ will converge to this equilibrium point. Otherwise, the equilibrium point is unstable and no convergence is observed. This topic is addressed in Examples 2 and 3. However, before exploring those, we will first examine the order of convergence for the approximate solutions and their first derivatives.

Example 1.In this example, we will employ the proposed HDG method to solve the Brusselator equation (27) with Dirichlet boundary conditions and coefficients η = 0 . 25 , ξ = 1 $$ \eta &#x0003D;0.25,\kern0.3em \xi &#x0003D;1 $$ , and γ = 0 $$ \gamma &#x0003D;0 $$ , that is, [34]

∂ ∂ t u 1 − 0 . 25 Δ u 1 − u 1 2 u 2 + 2 u 1 = r 1 , ∂ ∂ t u 2 − 0 . 25 Δ u 2 + u 1 2 u 2 − u 1 = r 2 , $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1-0.25\Delta {\mathfrak{u}}_1-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;2{\mathfrak{u}}_1&#x0003D;&amp; {\mathfrak{r}}_1,\hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2-0.25\Delta {\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0003D;&amp; {\mathfrak{r}}_2,\hfill \end{array}} $$ (29)
where Ω = ( 0 , 1 ) × ( 0 , 1 ) $$ \Omega &#x0003D;\left(0,1\right)\times \left(0,1\right) $$ . The exact solutions are given by u 1 = exp ( − x 1 − x 2 − t 2 ) $$ {\mathfrak{u}}_1&#x0003D;\exp \left(-{x}_1-{x}_2-\frac{t}{2}\right) $$ and u 2 = exp ( x 1 + x 2 + t 2 ) $$ {\mathfrak{u}}_2&#x0003D;\exp \left({x}_1&#x0002B;{x}_2&#x0002B;\frac{t}{2}\right) $$ . Therefore, the right-hand side functions r 1 $$ {\mathfrak{r}}_1 $$ and r 2 $$ {\mathfrak{r}}_2 $$ , as well as the initial and boundary conditions, can be derived easily using the exact solutions. The results are reported in Table 1 for approximate polynomials of degree k = 0 , 1 , 2 $$ k&#x0003D;0,1,2 $$ at final time T = 0 . 1 $$ T&#x0003D;0.1 $$ . As expected, the L 2 $$ {L}&#x0005E;2 $$ -error norms are satisfactory, demonstrating that the approximate solutions accurately estimate the analytical solutions. Also, in Table 1, we can observe the optimal order of convergence k + 1 $$ k&#x0002B;1 $$ in the spatial domain Ω $$ \Omega $$ for u 1 , u 2 $$ {u}_1,\kern0.3em {u}_2 $$ and their first-order derivatives. To achieve the desired order of convergence, when halving the spatial mesh size h $$ h $$ at each step, the time step size should be adjusted by dividing it by 2 k − 1 $$ {2}&#x0005E;{k-1} $$ . To this end, the first time step size is considered as δ t = 0 . 01 $$ \delta t&#x0003D;0.01 $$ . In Table 2, we present another test to evaluate the time convergence order of the method. As expected, we observe first-order convergence for both the approximate solutions and their first derivatives, which is attributed to the use of the Euler method for time discretization.

Additionally, we conducted another test to evaluate the accuracy of the proposed method at a longer final T = 4 $$ T&#x0003D;4 $$ . In Figure 2, the contour plot of the absolute value of pointwise errors between the numerical solutions and analytical solutions are illustrated for approximation polynomials of degree one, δ t = 0 . 005 $$ \delta t&#x0003D;0.005 $$ , and h = 0 . 1 $$ h&#x0003D;0.1 $$ . It is shown that the approximate solutions estimate the analytical solutions with high accuracy.

Details are in the caption following the image
FIGURE 2
Open in figure viewerPowerPoint
The contour plot of the absolute value of pointwise errors of numerical solutions and analytical solutions for Example 1 at final time T = 4 $$ T&#x0003D;4 $$ with τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ . [Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 1. L 2 $$ {L}&#x0005E;2 $$ error norms and corresponding spatial accuracy of the associated approximate solutions for Example 1 with polynomials of degree k = 0 , 1 , 2 $$ k&#x0003D;0,1,2 $$ and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ at final time T = 0 . 1 $$ T&#x0003D;0.1 $$ .
k $$ k $$ h $$ h $$ ‖ u 1 − u 1 ‖ Ω $$ {\left\Vert {u}_1-{\mathfrak{u}}_1\right\Vert}_{\Omega} $$ Spatial order ‖ q 1 − q 1 ‖ Ω $$ {\left\Vert {q}_1-{\mathfrak{q}}_1\right\Vert}_{\Omega} $$ Spatial order
0 0.2 2.8482 E-3 4.4793 E-2
0.1 1.5502 E-3 0.87 2.4915 E-2 0.85
0.05 8.3760 E-4 0.89 1.2562 E-2 0.99
1 0.4 5.5333 E-4 1.0097 E-2
0.2 1.2440 E-4 2.15 2.7227 E-3 1.89
0.1 3.2677 E-5 1.93 7.1223 E-4 1.94
2 0.4 6.4595 E-5 4.9854 E-4
0.2 7.3372 E-6 3.14 5.6062 E-5 3.15
0.1 8.8214 E-7 3.06 6.7099 E-6 3.06
k $$ k $$ h $$ h $$ ‖ u 2 − u 2 ‖ Ω $$ {\left\Vert {u}_2-{\mathfrak{u}}_2\right\Vert}_{\Omega} $$ Spatial order ‖ q 2 − q 2 ‖ Ω $$ {\left\Vert {q}_2-{\mathfrak{q}}_2\right\Vert}_{\Omega} $$ Spatial order
0 0.2 2.4321 E-2 3.7904 E-1
0.2 1.3360 E-2 0.86 3.6709 E-1 0.87
0.05 6.5237 E-3 1.03 1.0216 E-1 0.98
1 0.4 5.1940 E-3 5.4826 E-2
0.2 1.5904 E-3 1.71 1.4923 E-2 1.88
0.1 4.5656 E-4 1.80 3.7385 E-3 2.00
2 0.4 5.4681 E-4 4.3203 E-3
0.2 5.9730 E-5 3.19 4.8290 E-4 3.16
0.1 6.9986 E-6 3.09 5.8120 E-5 3.05
TABLE 2. L 2 $$ {L}&#x0005E;2 $$ error norms and corresponding temporal accuracy of the associated approximate solutions for Example 1 with polynomials of degree one and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ at final time T = 5 $$ T&#x0003D;5 $$ .
δ t $$ \delta t $$ ‖ u 1 − u 1 ‖ Ω $$ {\left\Vert {u}_1-{\mathfrak{u}}_1\right\Vert}_{\Omega} $$ Temporal order ‖ q 1 − q 1 ‖ Ω $$ {\left\Vert {q}_1-{\mathfrak{q}}_1\right\Vert}_{\Omega} $$ Temporal order
0.8 8.7476 E-4 4.0123 E-3
0.4 4.1218 E-4 1.09 1.8911 E-3 1.09
0.2 1.18081 E-4 1.19 8.3003 E-4 1.19
0.1 8.9022 E-5 1.02 4.0922 E-4 1.02
0.05 4.4423 E-5 1.01 2.0423 E-4 1.01
0.025 2.2114 E-5 1.00 1.0378 E-4 0.98
δ t $$ \delta t $$ ‖ u 2 − u 2 ‖ Ω $$ {\left\Vert {u}_2-{\mathfrak{u}}_2\right\Vert}_{\Omega} $$ Temporal order ‖ q 2 − q 2 ‖ Ω $$ {\left\Vert {q}_2-{\mathfrak{q}}_2\right\Vert}_{\Omega} $$ Temporal order
0.8 4.1741 E-1 1.9506
0.4 2.2220 E-1 0.91 1.03777 0.91
0.2 1.2684 E-1 0.81 5.9328 E-1 0.81
0.1 6.4439 E-2 0.98 3.0220 E-1 0.97
0.05 3.2529 E-2 0.99 1.5393 E-1 0.97
0.025 1.6405 E-2 0.99 8.0188 E-2 0.95

Example 2.Consider the Brusselator system (27) with imposing homogeneous Neumann boundary conditions and setting ξ = 0 . 5 , γ = 1 $$ \xi &#x0003D;0.5,\kern0.3em \gamma &#x0003D;1 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ ; see [33, 34]. Also, consider the following initial conditions:

u 1 , 0 = 1 2 x 1 2 − 1 3 x 1 3 , u 2 , 0 = 1 2 x 2 2 − 1 3 x 2 3 $$ {\mathfrak{u}}_{1,0}&#x0003D;\frac{1}{2}{x}_1&#x0005E;2-\frac{1}{3}{x}_1&#x0005E;3,\kern2em {\mathfrak{u}}_{2,0}&#x0003D;\frac{1}{2}{x}_2&#x0005E;2-\frac{1}{3}{x}_2&#x0005E;3 $$ (30)
where ( x 1 , x 2 ) ∈ Ω = ( 0 , 1 ) × ( 0 , 1 ) $$ \left({x}_1,{x}_2\right)\in \Omega &#x0003D;\left(0,1\right)\times \left(0,1\right) $$ and t ∈ ( 0 , 10 ] $$ t\in \left(0,10\right] $$ . The approximations of u 1 $$ {u}_1 $$ and u 2 $$ {u}_2 $$ are reported in Table 3 at points ( 0 . 2 , 0 . 2 , t ) , ( 0 . 4 , 0 . 6 , t ) , ( 0 . 5 , 0 . 5 , t ) $$ \left(0.2,0.2,t\right),\kern0.3em \left(0.4,0.6,t\right),\kern0.3em \left(0.5,0.5,t\right) $$ , and ( 0 . 8 , 0 . 9 , t ) $$ \left(0.8,0.9,t\right) $$ for t = 1 , 2 , … , 10 $$ t&#x0003D;1,2,\dots, 10 $$ with approximate polynomial of degree one and δ t = 0 . 005 $$ \delta t&#x0003D;0.005 $$ . Due to 1 − ξ + γ 2 > 0 $$ 1-\xi &#x0002B;{\gamma}&#x0005E;2&gt;0 $$ , the approximate solutions ( u 1 , u 2 ) $$ \left({u}_1,{u}_2\right) $$ converge to critical point 1 , 1 2 $$ \left(1,\frac{1}{2}\right) $$ . Moreover, in Figure 3, by showing u 1 $$ {u}_1 $$ and u 2 $$ {u}_2 $$ at times t = 2 , 4 , 6 , 8 , 10 $$ t&#x0003D;2,4,6,8,10 $$ , the convergence can be seen in the spatial domain by concentrating on the range of color data.

For further investigation, again consider (27) with homogeneous Neumann boundary conditions and initial data

u 1 , 0 = 2 + 0 . 25 x 2 , u 2 , 0 = 1 + 0 . 8 x 1 $$ {\mathfrak{u}}_{1,0}&#x0003D;2&#x0002B;0.25{x}_2,\kern2em {\mathfrak{u}}_{2,0}&#x0003D;1&#x0002B;0.8{x}_1 $$ (31)
where ( x 1 , x 2 ) ∈ Ω = ( 0 , 1 ) × ( 0 , 1 ) $$ \left({x}_1,{x}_2\right)\in \Omega &#x0003D;\left(0,1\right)\times \left(0,1\right) $$ and t ∈ ( 0 , 10 ] $$ t\in \left(0,10\right] $$ . Moreover, the coefficients are chosen as ξ = 1 , γ = 2 $$ \xi &#x0003D;1,\kern0.3em \gamma &#x0003D;2 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ where 1 − ξ + γ 2 > 0 $$ 1-\xi &#x0002B;{\gamma}&#x0005E;2&gt;0 $$ . The reported results in Table 4 show the convergence of ( u 1 , u 2 ) $$ \left({u}_1,{u}_2\right) $$ to equilibrium point ( 2 , 0 . 5 ) $$ \left(2,0.5\right) $$ at different points ( 0 . 2 , 0 . 2 , t ) , ( 0 . 4 , 0 . 6 , t ) , ( 0 . 5 , 0 . 5 , t ) $$ \left(0.2,0.2,t\right),\kern0.3em \left(0.4,0.6,t\right),\kern0.3em \left(0.5,0.5,t\right) $$ , and ( 0 . 8 , 0 . 9 , t ) $$ \left(0.8,0.9,t\right) $$ , for t = 1 , 2 , … , 10 $$ t&#x0003D;1,2,\dots, 10 $$ . Due to the linearity of the initial conditions (31), the faster convergence is observed in Table 4 in comparison to Table 3.

Next, we investigate the order of convergence using the following formula:

O j ( u ) = log 2 ‖ u h / 2 j − u h / j ‖ 2 ‖ u h / 4 j − u h / 2 j ‖ 2 $$ {\mathrm{O}}_j(u)&#x0003D;{\log}_2\left(\frac{{\left\Vert {u}_{h/2j}-{u}_{h/j}\right\Vert}_2}{{\left\Vert {u}_{h/4j}-{u}_{h/2j}\right\Vert}_2}\right) $$ (32)
where u h / j , u h / 2 j $$ {u}_{h/j},\kern0.3em {u}_{h/2j} $$ , and u h / 4 j $$ {u}_{h/4j} $$ represent the approximate solutions with spatial mesh sizes h / j , h / 2 j $$ h/j,\kern0.3em h/2j $$ , and h / 4 j $$ h/4j $$ , respectively. As shown in Table 5, we achieve the optimal convergence order for polynomial degrees k = 0 , 1 $$ k&#x0003D;0,1 $$ , with j = 1 , … , 5 $$ j&#x0003D;1,\dots, 5 $$ , and h = 0 . 8 $$ h&#x0003D;0.8 $$ .

TABLE 3. The values of the associated approximate solutions for Example 2, with initial conditions (30) at selected different points. The approximate polynomials are degree one, h = 0 . 1 , δ t = 0 . 005 $$ h&#x0003D;0.1,\kern0.3em \delta t&#x0003D;0.005 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ .
(0.2, 0.2) (0.4, 0.6) (0.5, 0.5) (0.8, 0.9)
t $$ t $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$
1 0.5313 0.1696 0.5496 0.2552 0.5539 0.2387 0.5804 0.3177
2 07032 0.3720 0.7274 0.4291 0.7253 0.4186 0.7529 0.4685
3 0.8182 0.4952 0.8456 0.5213 0.8415 0.5164 0.8696 0.5361
4 0.9108 0.5365 0.9336 0.5406 0.9297 0.5398 0.9509 0.5405
5 0.9721 0.5309 0.9849 0.5267 0.9827 0.5274 0.9934 0.5225
6 0.9999 0.5146 1.0043 0.5105 1.0035 0.5112 1.0066 0.5075
7 1.0064 0.5038 1.0067 0.5018 1.0066 0.5021 1.0065 0.5005
8 1.0047 0.4996 1.0040 0.4990 1.0042 0.4991 1.0034 0.4987
9 1.0021 0.4988 1.0015 0.4988 1.0016 0.4988 1.0011 0.4989
10 1.0005 0.4992 1.0002 0.4994 1.0003 0.4993 1.0001 0.4995
↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$
∞ $$ \infty $$ 1 0.5 1 0.5 1 0.5 1 0.5
Details are in the caption following the image
FIGURE 3
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Obtained approximate solutions u 1 $$ {u}_1 $$ (left) and u 2 $$ {u}_2 $$ (right) for Example 2 at times t = 2 , 4 , 6 , 8 , 10 $$ t&#x0003D;2,4,6,8,10 $$ (from top to bottom) with polynomial of degree one, δ t = 0 . 005 $$ \delta t&#x0003D;0.005 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ . [Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 4. The values of the associated approximate solutions for Example 2, with initial conditions (31) at selected different points. The approximate polynomials are degree one, δ t = 0 . 005 , h = 0 . 1 $$ \delta t&#x0003D;0.005,\kern0.3em h&#x0003D;0.1 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ .
(0.2, 0.2) (0.4, 0.6) (0.5, 0.5) (0.8, 0.9)
t $$ t $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$
1 2.3430 0.4167 2.4466 0.3954 2.4738 0.3906 2.6070 0.3684
2 2.0950 0.4686 2.1258 0.4600 2.1343 0.4577 2.1762 0.4471
3 2.0219 0.4915 2.0298 0.4887 2.0320 0.4880 2.0433 0.4842
4 2.0044 0.4981 2.0061 0.4975 2.0066 0.4973 2.0091 0.4963
5 2.0008 0.4996 2.0011 0.4995 2.0012 0.4995 2.0017 0.4993
6 2.0001 0.4999 2.0002 0.4999 2.0002 0.4999 2.0003 0.4999
7 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000
8 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000
9 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000
10 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000 2.0000 0.5000
↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$
∞ $$ \infty $$ 2 0.5 2 0.5 2 0.5 2 0.5
TABLE 5. The evaluated order of convergence in the spatial domain using the relation (32) for Example 2. The results are presented for polynomial degrees k = 0 , 1 $$ k&#x0003D;0,1 $$ and stabilization parameters τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ at final time T = 10 $$ T&#x0003D;10 $$ .
k $$ k $$ j $$ j $$ O j ( u 1 ) $$ {O}_j\left({u}_1\right) $$ O j ( q 1 ) $$ {O}_j\left({q}_1\right) $$ O j ( u 2 ) $$ {O}_j\left({u}_2\right) $$ O j ( q 2 ) $$ {O}_j\left({q}_2\right) $$
0 1 0.26 0.18 0.19 0.09
2 0.25 0.19 0.20 0.10
3 0.54 0.49 0.60 0.46
4 0.85 0.86 0.83 0.72
5 0.95 0.88 0.91 0.83
Expected order 1 1 1 1
1 1 1.32 0.44 1.16 0.39
2 1.88 1.65 1.79 1.72
3 1.90 1.71 1.85 1.89
4 1.92 1.89 1.93 1.90
Expected order 2 2 2 2

Example 3.In this numerical example, we intend to examine Brusselator system with stable and unstable equilibrium points. Consider Brusselator system (27) with homogeneous Neumann boundary conditions and initial conditions u 1 , 0 = 0 . 5 + x 2 $$ {\mathfrak{u}}_{1,0}&#x0003D;0.5&#x0002B;{x}_2 $$ and u 2 , 0 = 1 + 5 x 1 $$ {\mathfrak{u}}_{2,0}&#x0003D;1&#x0002B;5{x}_1 $$ where ( x 1 , x 2 ) ∈ Ω = [ 0 , 1 ] × [ 0 , 1 ] $$ \left({x}_1,{x}_2\right)\in \Omega &#x0003D;\left[0,1\right]\times \left[0,1\right] $$ ; see [32, 33]. In Figure 4, the results are reported for ξ = 1 , γ = 3 . 4 $$ \xi &#x0003D;1,\kern0.3em \gamma &#x0003D;3.4 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ . Indeed 1 − ξ + γ 2 = ( 3 . 4 ) 2 > 0 $$ 1-\xi &#x0002B;{\gamma}&#x0005E;2&#x0003D;{(3.4)}&#x0005E;2&gt;0 $$ and the approximate solutions converge to the stable equilibrium point ( 3 . 4 , 1 / 3 . 4 ) $$ \left(3.4,1/3.4\right) $$ . In contrast, by choosing ξ = 3 . 4 , γ = 1 $$ \xi &#x0003D;3.4,\kern0.3em \gamma &#x0003D;1 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ , we get 1 − ξ + γ 2 = 2 − ( 3 . 4 ) 2 < 0 $$ 1-\xi &#x0002B;{\gamma}&#x0005E;2&#x0003D;2-{(3.4)}&#x0005E;2&lt;0 $$ , and theoretically, we expect that the solution does not exhibit any convergence behavior. As seen, the HDG method produces divergence approximation solutions according to the theoretical achievements for unstable equilibrium point; see Figure 5.

Details are in the caption following the image
FIGURE 4
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Obtained approximate solutions u 1 $$ {u}_1 $$ (the left side) and u 2 $$ {u}_2 $$ (the right side) for Example 3 with ξ = 1 , γ = 3 . 4 $$ \xi &#x0003D;1,\kern0.3em \gamma &#x0003D;3.4 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ at times t = 0 , 2 , 4 , 7 , 10 , 20 $$ t&#x0003D;0,2,4,7,10,20 $$ . The approximate polynomials are degree one, h = 0 . 1 , δ t = 0 . 05 $$ h&#x0003D;0.1,\kern0.3em \delta t&#x0003D;0.05 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ . [Colour figure can be viewed at wileyonlinelibrary.com]
Details are in the caption following the image
FIGURE 5
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Obtained approximate solutions u 1 $$ {u}_1 $$ (the left side) and u 2 $$ {u}_2 $$ (the right side) for Example 3 with ξ = 3 . 4 , γ = 1 $$ \xi &#x0003D;3.4,\kern0.3em \gamma &#x0003D;1 $$ , and η = 0 . 002 $$ \eta &#x0003D;0.002 $$ at times t = 0 , 2 , 4 , 7 , 10 , 20 $$ t&#x0003D;0,2,4,7,10,20 $$ . The approximate polynomials are degree one, h = 0 . 1 , δ t = 0 . 05 $$ h&#x0003D;0.1,\kern0.3em \delta t&#x0003D;0.05 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ . [Colour figure can be viewed at wileyonlinelibrary.com]

4.2 Glycolysis System

By considering (1) with m = 2 , f 1 ( u 1 , u 2 ) = μ u 2 − u 1 + u 1 2 u 2 , f 2 ( u 1 , u 2 ) = κ − μ u 2 − u 1 2 u 2 $$ m&#x0003D;2,\kern0.3em {\mathfrak{f}}_1\left({\mathfrak{u}}_1,{\mathfrak{u}}_2\right)&#x0003D;\mu {\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0002B;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2,\kern0.3em {\mathfrak{f}}_2\left({\mathfrak{u}}_1,{\mathfrak{u}}_2\right)&#x0003D;\kappa -\mu {\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2 $$ , and η 1 = η 2 = η $$ {\eta}_1&#x0003D;{\eta}_2&#x0003D;\eta $$ , the following glycolysis reaction–diffusion system [32] is achieved
∂ ∂ t u 1 = η Δ u 1 + μ u 2 − u 1 + u 1 2 u 2 , ∂ ∂ t u 2 = η Δ u 2 + κ − μ u 2 − u 1 2 u 2 , $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1&amp; &#x0003D;\eta \Delta {\mathfrak{u}}_1&#x0002B;\mu {\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0002B;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2,\hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2&amp; &#x0003D;\eta \Delta {\mathfrak{u}}_2&#x0002B;\kappa -\mu {\mathfrak{u}}_2-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2,\hfill \end{array}} $$ (33)
where μ $$ \mu $$ and κ $$ \kappa $$ are real value constants. Similar to the Brusselator system, for small values of η $$ \eta $$ , the dissipative terms in (33) become negligible, reducing system (33) to a system of ODEs. Under the condition
κ 4 + ( 2 μ − 1 ) κ 2 + ( κ + κ 2 ) μ + κ 2 < 0 $$ \frac{\kappa&#x0005E;4&#x0002B;\left(2\mu -1\right){\kappa}&#x0005E;2&#x0002B;\left(\kappa &#x0002B;{\kappa}&#x0005E;2\right)}{\mu &#x0002B;{\kappa}&#x0005E;2}&lt;0 $$ (34)
the equilibrium point κ , κ μ + κ 2 $$ \left(\kappa, \frac{\kappa }{\mu &#x0002B;{\kappa}&#x0005E;2}\right) $$ is stable and approximate solutions ( u 1 , u 2 ) $$ \left({u}_1,{u}_2\right) $$ converge to this point as t $$ t $$ goes to infinity [35, 41]. Otherwise, the equilibrium point is unstable. In the following, we demonstrate that this stability property is preserved by the proposed HDG method.

Example 4.Consider the following glycolysis system which is subjected to Dirichlet boundary conditions:

∂ ∂ t u 1 − 0 . 25 Δ u 1 − u 1 2 u 2 + u 1 − u 2 = r 1 , ∂ ∂ t u 2 − 0 . 25 Δ u 2 + u 1 2 u 2 + u 2 = r 2 , $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1&amp; -0.25\Delta {\mathfrak{u}}_1-{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_1-{\mathfrak{u}}_2&#x0003D;{\mathfrak{r}}_1,\hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2&amp; -0.25\Delta {\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_1&#x0005E;2{\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_2&#x0003D;{\mathfrak{r}}_2,\hfill \end{array}} $$
where Ω = [ 0 , 1 ] × [ 0 , 1 ] $$ \Omega &#x0003D;\left[0,1\right]\times \left[0,1\right] $$ , and the right-hand side functions r 1 $$ {\mathfrak{r}}_1 $$ and r 2 $$ {\mathfrak{r}}_2 $$ are obtained by exact solutions u 1 = exp − t 2 sin ( − x 1 − x 2 ) $$ {\mathfrak{u}}_1&#x0003D;\exp \left(-\frac{t}{2}\right)\sin \left(-{x}_1-{x}_2\right) $$ and u 2 = exp − t 2 cos ( x 1 + x 2 ) $$ {\mathfrak{u}}_2&#x0003D;\exp \left(-\frac{t}{2}\right)\cos \left({x}_1&#x0002B;{x}_2\right) $$ . The L 2 $$ {L}&#x0005E;2 $$ error norms and corresponding order of accuracy are given in Table 6, for approximate polynomials of degree k = 0 , 1 , 2 $$ k&#x0003D;0,1,2 $$ at final time T = 0 . 1 $$ T&#x0003D;0.1 $$ . In this test, the initial time step size is set to δ t = 0 . 01 $$ \delta t&#x0003D;0.01 $$ , and with each halving of the spatial mesh size h , δ t $$ h,\kern0.3em \delta t $$ is updated by dividing it by 2 k − 1 $$ {2}&#x0005E;{k-1} $$ . As seen, the results are satisfactory, and the optimal order of convergence is gained in the spatial domain Ω $$ \Omega $$ for u 1 , u 2 $$ {u}_1,\kern0.3em {u}_2 $$ and their first-order partial derivatives.

TABLE 6. L 2 $$ {L}&#x0005E;2 $$ error norms and corresponding spatial accuracy of the associated approximate solutions for Example 4 with approximate polynomials of degree k = 0 , 1 , 2 $$ k&#x0003D;0,1,2 $$ and τ 1 = τ 2 = − 1 . 5 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1.5 $$ at final time T = 0 . 1 $$ T&#x0003D;0.1 $$ .
k $$ k $$ h $$ h $$ ‖ u 1 − u 1 ‖ Ω $$ {\left\Vert {u}_1-{\mathfrak{u}}_1\right\Vert}_{\Omega} $$ Order ‖ q 1 − q 1 ‖ Ω $$ {\left\Vert {q}_1-{\mathfrak{q}}_1\right\Vert}_{\Omega} $$ Order
0 0.2 2.4452 E-3 3.7635 E-2
0.1 1.3375 E-3 0.88 1.9306 E-2 0.96
0.05 7.3079 E-4 0.88 9.8531 E-3 0.97
1 0.2 1.9501 E-4 6.6192 E-3
0.1 3.8494 E-5 2.34 1.7486 E-3 1.92
0.05 9.0851 E-6 2.08 4.4117 E-4 1.99
2 0.2 1.2422 E-4 9.8089 E-4
0.1 1.3110 E-5 3.24 1.0141 E-4 3.27
0.05 1.4484 E-6 3.18 1.1180 E-5 3.18
k $$ k $$ h $$ h $$ ‖ u 2 − u 2 ‖ Ω $$ {\left\Vert {u}_2-{\mathfrak{u}}_2\right\Vert}_{\Omega} $$ Order ‖ q 2 − q 2 ‖ Ω $$ {\left\Vert {q}_2-{\mathfrak{q}}_2\right\Vert}_{\Omega} $$ Order
0 0.2 2.1956 E-3 3.6199 E-2
0.1 1.1486 E-3 0.93 1.9115 E-2 0.92
0.05 5.9749 E-4 0.94 9.6999 E-3 0.98
1 0.2 1.8493 E-4 4.9198 E-3
0.1 4.5424 E-5 2.03 1.3165 E-3 1.90
0.05 1.1069 E-5 2.04 3.3108 E-4 1.99
2 0.2 1.5340 E-4 1.1535 E-3
0.1 1.6412 E-5 3.22 1.3508 E-4 3.09
0.05 1.8499 E-6 3.15 1.6044 E-5 3.07

Example 5.Consider system (33) with Neumann boundary conditions and initial conditions u 1 , 0 = 1 + x 1 + 0 . 5 x 2 $$ {\mathfrak{u}}_{1,0}&#x0003D;1&#x0002B;{x}_1&#x0002B;0.5{x}_2 $$ and u 2 , 0 = 1 / 3 $$ {\mathfrak{u}}_{2,0}&#x0003D;1/3 $$ , as presented in [32]. The values of approximate solution ( u 1 , u 2 ) $$ \left({u}_1,{u}_2\right) $$ are reported in Table 7 at the points ( 0 . 2 , 0 . 2 ) , ( 0 . 4 , 0 . 6 ) , ( 0 . 5 , 0 . 5 ) $$ \left(0.2,0.2\right),\kern0.3em \left(0.4,0.6\right),\kern0.3em \left(0.5,0.5\right) $$ , and ( 0 . 8 , 0 . 9 ) $$ \left(0.8,0.9\right) $$ for μ = 3 . 5 , κ = 0 . 25 $$ \mu &#x0003D;3.5,\kern0.3em \kappa &#x0003D;0.25 $$ , and η = 0 . 001 $$ \eta &#x0003D;0.001 $$ . As observed, the approximate solution converges to equilibrium point ( 0 . 25 , 0 . 25 3 . 5 + ( 0 . 25 ) 2 ) $$ \left(0.25,\frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2}\right) $$ , as the condition in (34) is satisfied, making ( 0 . 25 , 0 . 25 3 . 5 + ( 0 . 25 ) 2 ) $$ \left(0.25,\frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2}\right) $$ a stable equilibrium point. In contrast, by setting μ = 0 . 008 , κ = 0 . 6 $$ \mu &#x0003D;0.008,\kern0.3em \kappa &#x0003D;0.6 $$ , and η = 0 . 001 $$ \eta &#x0003D;0.001 $$ , the system yields an unstable equilibrium point. Therefore, as we expected, no convergence is observed in Figure 6.

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FIGURE 6
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Obtained approximate solutions u 1 $$ {u}_1 $$ (the left side) and u 2 $$ {u}_2 $$ (the right side) for Example 5 with μ = 0 . 008 , κ = 0 . 6 $$ \mu &#x0003D;0.008,\kern0.3em \kappa &#x0003D;0.6 $$ , and η = 0 . 001 $$ \eta &#x0003D;0.001 $$ at times t = 0 , 4 , 8 , 12 , 16 , 20 $$ t&#x0003D;0,4,8,12,16,20 $$ . The approximate polynomials are degree one, h = 0 . 1 , δ t = 0 . 05 $$ h&#x0003D;0.1,\kern0.3em \delta t&#x0003D;0.05 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ . [Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 7. The values of the associated approximate solutions for Example 5, with μ = 3 . 5 , κ = 0 . 25 $$ \mu &#x0003D;3.5,\kern0.3em \kappa &#x0003D;0.25 $$ , and η = 0 . 001 $$ \eta &#x0003D;0.001 $$ . The approximate polynomials are degree one, δ t = 0 . 005 , h = 0 . 1 $$ \delta t&#x0003D;0.005,\kern0.3em h&#x0003D;0.1 $$ , and τ 1 = τ 2 = − 1 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;-1 $$ .
(0.2, 0.2) (0.4, 0.6) (0.5, 0.5) (0.8, 0.9)
t $$ t $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$ u 1 $$ {u}_1 $$ u 2 $$ {u}_2 $$
2 0.4426 0.0662 0.4979 0.0647 0.5049 0.0645 0.5759 0.0624
4 0.2763 0.0698 0.2837 0.0696 0.2846 0.0696 0.2940 0.0695
6 0.2536 0.0701 0.2546 0.0701 0.2548 0.0701 0.2560 0.0701
8 0.2505 0.0702 0.2506 0.0702 0.2507 0.0702 0.2508 0.0702
10 0.2501 0.0702 0.2501 0.0702 0.2501 0.0702 0.2501 0.0702
12 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702
14 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702
16 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702
18 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702
20 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702 0.2500 0.0702
↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$ ↓ $$ \downarrow $$
∞ $$ \infty $$ 0.25 0 . 25 3 . 5 + ( 0 . 25 ) 2 $$ \frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2} $$ 0.25 0 . 25 3 . 5 + ( 0 . 25 ) 2 $$ \frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2} $$ 0.25 0 . 25 3 . 5 + ( 0 . 25 ) 2 $$ \frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2} $$ 0.25 0 . 25 3 . 5 + ( 0 . 25 ) 2 $$ \frac{0.25}{3.5&#x0002B;{(0.25)}&#x0005E;2} $$

Example 6.By setting m = 3 , f 1 ( u 1 , u 2 , u 3 ) = − u 1 u 2 + u 3 , f 2 ( u 1 , u 2 , u 3 ) = − u 1 u 2 + u 3 $$ m&#x0003D;3,\kern0.3em {\mathfrak{f}}_1\left({\mathfrak{u}}_1,{\mathfrak{u}}_2,{\mathfrak{u}}_3\right)&#x0003D;-{\mathfrak{u}}_1{\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_3,\kern0.3em {\mathfrak{f}}_2\left({\mathfrak{u}}_1,{\mathfrak{u}}_2,{\mathfrak{u}}_3\right)&#x0003D;-{\mathfrak{u}}_1{\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_3 $$ , and f 3 ( u 1 , u 2 , u 3 ) $$ {\mathfrak{f}}_3\left({\mathfrak{u}}_1,{\mathfrak{u}}_2,{\mathfrak{u}}_3\right) $$ = u 1 u 2 − u 3 $$ &#x0003D;{\mathfrak{u}}_1{\mathfrak{u}}_2-{\mathfrak{u}}_3 $$ in system (1), the following three-component system is obtained.

∂ ∂ t u 1 − η 1 Δ u 1 + u 1 u 2 − u 3 = 0 , ∂ ∂ t u 2 − η 2 Δ u 2 + u 1 u 2 − u 3 = 0 , ∂ ∂ t u 3 − η 3 Δ u 3 − u 1 u 2 + u 3 = 0 , $$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_1-{\eta}_1\Delta {\mathfrak{u}}_1&#x0002B;{\mathfrak{u}}_1{\mathfrak{u}}_2-{\mathfrak{u}}_3&#x0003D;&amp; 0,\hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_2-{\eta}_2\Delta {\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_1{\mathfrak{u}}_2-{\mathfrak{u}}_3&#x0003D;&amp; 0,\hfill \\ {}\hfill \frac{\partial }{\partial t}{\mathfrak{u}}_3-{\eta}_3\Delta {\mathfrak{u}}_3-{\mathfrak{u}}_1{\mathfrak{u}}_2&#x0002B;{\mathfrak{u}}_3&#x0003D;&amp; 0,\hfill \end{array}} $$ (35)
which is equipped with Neumann boundary conditions; see [29]. In Figure 7, the approximate solutions are plotted for η 1 = 0 . 05 , η 2 = 1 , η 3 = 0 . 1 $$ {\eta}_1&#x0003D;0.05,{\eta}_2&#x0003D;1,\kern0.3em {\eta}_3&#x0003D;0.1 $$ , and the following initial conditions
u 1 , 0 = 1 2 − tanh x 2 + y 2 − 0 . 2 0 . 1 + 1 + 0 . 01 , u 2 , 0 = 1 2 − tanh x 2 + y 2 − 0 . 2 0 . 1 + 1 + 0 . 01 , u 3 , 0 = 1 4 tanh x 2 + ( y − 0 . 2 ) 2 − 0 . 2 0 . 1 + 1 + 1 4 tanh x 2 + ( y + 0 . 2 ) 2 − 0 . 2 0 . 1 + 1 + 0 . 01 . $$ {\displaystyle \begin{array}{cc}\hfill {\mathfrak{u}}_{1,0}&amp; &#x0003D;\frac{1}{2}\left(-\tanh \left(\frac{\sqrt{x&#x0005E;2&#x0002B;{y}&#x0005E;2}-0.2}{0.1}\right)&#x0002B;1\right)&#x0002B;0.01,\hfill \\ {}\hfill {\mathfrak{u}}_{2,0}&amp; &#x0003D;\frac{1}{2}\left(-\tanh \left(\frac{\sqrt{x&#x0005E;2&#x0002B;{y}&#x0005E;2}-0.2}{0.1}\right)&#x0002B;1\right)&#x0002B;0.01,\hfill \\ {}\hfill {\mathfrak{u}}_{3,0}&amp; &#x0003D;\frac{1}{4}\left(\tanh \left(\frac{\sqrt{x&#x0005E;2&#x0002B;{\left(y-0.2\right)}&#x0005E;2}-0.2}{0.1}\right)&#x0002B;1\right)\hfill \\ {}\hfill &amp; \kern10pt &#x0002B;\frac{1}{4}\left(\tanh \left(\frac{\sqrt{x&#x0005E;2&#x0002B;{\left(y&#x0002B;0.2\right)}&#x0005E;2}-0.2}{0.1}\right)&#x0002B;1\right)&#x0002B;0.01.\hfill \end{array}} $$
The results are obtained using approximate polynomials of degree one, with δ t = 0 . 005 , h = 0 . 2 $$ \delta t&#x0003D;0.005,\kern0.3em h&#x0003D;0.2 $$ , and τ 1 = τ 2 = τ 3 = − 20 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;{\tau}_3&#x0003D;-20 $$ . Compared to [29] and by focusing on the color data range, it is evident that system (35) reaches the true constant steady state, demonstrating the capability of the proposed HDG method to provide accurate and acceptable approximate solutions.

Details are in the caption following the image
FIGURE 7
Open in figure viewerPowerPoint
Obtained approximate solutions u 1 $$ {u}_1 $$ (the left side), u 2 $$ {u}_2 $$ (in the middle), and u 3 $$ {u}_3 $$ (the right side) for Example 6 at times t = 0 , 0 . 2 , 1 , 2 $$ t&#x0003D;0,0.2,1,2 $$ . The approximate polynomials are degree one, h = 0 . 2 , δ t = 0 . 005 $$ h&#x0003D;0.2,\kern0.3em \delta t&#x0003D;0.005 $$ , and τ 1 = τ 2 = τ 3 = − 20 $$ {\tau}_1&#x0003D;{\tau}_2&#x0003D;{\tau}_3&#x0003D;-20 $$ . [Colour figure can be viewed at wileyonlinelibrary.com. [Colour figure can be viewed at wileyonlinelibrary.com]

5 Conclusion

In this paper, a stable HDG method has been proposed for solving the 2D nonlinear m $$ m $$ -component reaction–diffusion system. The semidiscrete scheme is obtained by using the backward Euler method for time discretization. The fully discrete scheme has been achieved by applying an HDG method for the spatial domain. A suitable definition of numerical fluxes and traces is the success key of the HDG method which is led it to the numerically stable method. In Theorem 1, we have proved that by considering homogeneous Dirichlet boundary conditions and under some mild conditions on the stabilization parameters, the method is numerically stable in the sense of the energy method. In Corollary 1, the stability theorem has been extended for problems with homogeneous Neumann boundary conditions. Due to the existence of nonlinear terms in system (1), a section is devoted to presenting a linearization method. Therefore, a linear system has been presented by introducing some bilinear and functional terms. To get lower computational costs, we offer Schur's complement idea for solving the linear system. Sufficient numerical results are provided for confirming the stability of the proposed method by applying the HDG method to a three-component system, Brusselator system, and glycolysis system. For both systems, the HDG method demonstrated the optimal rate of convergence. Indeed, for a generated mesh with approximation polynomials of order, k $$ k $$ in each element, approximate solutions, and their first partial derivatives converge to their corresponding exact values with order k + 1 $$ k&#x0002B;1 $$ . Selected physical properties of Brusselator and glycolysis systems are discussed in numerical examples. Approximate solutions have shown convergence behavior to the stable equilibrium points of Brusselator and glycolysis systems, and conversely, no convergence is observed to the unstable equilibrium points although in both cases the solution converge to a certain solution. The HDG method is one of the successful methods that can be constructed for such systems of differential equations, especially in higher dimensions.

Author Contributions

Shima Baharlouei: conceptualization, writing – original draft, writing – review and editing, investigation, visualization, validation, methodology, software, formal analysis. Reza Mokhtari: writing – review and editing, investigation, supervision, project administration. Nabi Chegini: writing – review and editing; supervision.

Conflicts of Interest

The authors declare no conflicts of interest.

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