This paper is concerned with stability analysis of additive Runge-Kutta methods for delay-integro-differential equations. We show that if the additive Runge-Kutta methods are algebraically stable, the perturbations of the numerical solutions are controlled by the initial perturbations from the system and the methods.

1. Introduction

Spatial discretization of many nonlinear parabolic problems usually gives a class of ordinary differential equations, which have the stiff part and the nonstiff part; see, e.g., [1–5]. In such cases, the most widely used time-discretizations are the special organized numerical methods, such as the implicit-explicit numerical methods [6, 7], the additive Runge-Kutta methods [8–12], and the linearized methods [13, 14]. When applying the split numerical methods to numerically solve the equations, it is important to investigate the stability of the numerical methods.

In this paper, it is assumed that the spatial discretization of time-dependent partial differential equations yields the following nonlinear delay-integro-differential equations:

()

Here τ is a positive delay term, ψ(t) is continuous, f^[1]: [t₀, +∞] × X → X, and f^[2]: [t₀, +∞] × X × X × X → X, such that problem (1) owns a unique solution, where X is a real or complex Hilbert space. Particularly, when g ≡ 0, problem (1) is reduced to the nonlinear delay differential equations. When the delay term τ = 0, problem (1) is reduced to the ordinary differential equations.

The investigation on stability analysis of different numerical methods for problem (1) has fascinated generations of researchers. For example, Torelli [15] considered stability of Euler methods for the nonautonomous nonlinear delay differential equations. Hout [16] studied the stability of Runge-Kutta methods for systems of delay differential equations. Baker and Ford [17] discussed stability of continuous Runge-Kutta methods for integrodifferential systems with unbounded delays. Zhang and Vandewalle [18] discussed the stability of the general linear methods for integrodifferential equations with memory. Li and Zhang obtained the stability and convergence of the discontinuous Galerkin methods for nonlinear delay differential equations [19, 20]. More references for this topic can be found in [21–30]. However, few works have been found on the stability of splitting methods for the proposed methods.

In the present work, we present the additive Runge-Kutta methods with some appropriate quadrature rules to numerically solve the nonlinear delay-integrodifferential equations (1). It is shown that if the additive Runge-Kutta methods are algebraically stable, the obtained numerical solutions are globally and asymptotically stable under the given assumptions, respectively. The rest of the paper is organized as follows. In Section 2, we present the numerical methods for problems (1). In Section 3, we consider stability analysis of the numerical schemes. Finally, we present some extensions in Section 4.

2. The Numerical Methods

In this section, we present the additive Runge-Kutta methods with the appropriate quadrature rules to numerically solve problem (1).

The coefficients of the additive Runge-Kutta methods can be organized in Buther tableau as follows (cf. [31]):

()

where , , and for k = 1,2.

Then, the presented ARKMs for problem (1) can be written by

()

where y_n and

are approximations to the analytic solution y(t_n) and y(t_n + c_ih), respectively, y_n = ψ(t_n) for n ≤ 0,

for t_n + c_ih ≤ 0, and

denotes the approximation to

, which can be computed by some appropriate quadrature rules

()

For example, we usually adopt the repeated Simpson’s rule or Newton-Cotes rule, etc., according to the requirement of the convergence of the method (cf. [18]).

3. Stability Analysis

In this section, we consider the numerical stability of the proposed methods. First, we introduce a perturbed problem, whose solution satisfies

()

It is assumed that there exist some inner product <·, ·> and the induced norm ‖·‖ such that

()

where α < 0, β₁ < 0, β₂ > 0, γ > 0, and θ > 0 are constants. It is remarkable that the conditions can be equivalent to the assumptions in [32, 33] (see. [34] Remark 2.1).

Definition 1 (cf. [9]). An additive Runge-Kutta method is called algebraically stable if the matrices

()

are nonnegative.

Theorem 2. Assume an additive Runge-Kutta method is algebraically stable and β₁ + β₂ + 4γτ²η²θ² < 0, where η = max⁡{p₁, p₂.⋯, p_k}. Then, it holds that

()

where y_n and z_n are numerical approximations to problems (1) and (5), respectively.

Proof. Let and be two sequences of approximations to problems (1) and (5), respectively, by ARKMs with the same stepsize h and write

()

With the notation, the ARKMs for (1) and (5) yield

()

Thus, we have

()

Since that the matrix

is a nonnegative matrix, we obtain

()

Furthermore, by conditions (6), we find

()

and

()

Together with (11), (12), (13), and (14), we get

()

Note that

()

Then, we obtain

()

Hence,

()

where

. This completes the proof.

Theorem 3. Assume an additive Runge-Kutta method is algebraically stable and β₁ + β₂ + 4γτ²η²θ² < 0. Then, it holds that

()

Proof. Similar to the proof of Theorem 2, it holds that

()

Note that β₁ + β₂ + 4γτ²η²θ² < 0 and

; we have

()

On the other hand,

()

Now, in view of (10), (21), (22), and (23), we obtain

()

This completes the proof.

Remark 4. In [35], Yuan et al. also discussed nonlinear stability of additive Runge-Kutta methods for multidelay-integro-differential equations. However, the main results are different. The main reason is that the results in [35] imply that the perturbations of the numerical solutions tend to infinity when the time increase, while the stability results in present paper indicate that the perturbations of the numerical solutions are independent of the time. Besides, the asymptotical stability of the methods is also discussed in the present paper.

4. Conclusion

The additive Runge-Kutta methods with some appropriate quadrature rules are applied to solve the delay-integro-differential equations. It is shown that if the additive Runge-Kutta methods are algebraically stable, the obtained numerical solutions can be globally and asymptotically stable, respectively. In the future works, we will apply the methods to solve more real-world problems.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (71601125).

Open Research

Data Availability

No data were used to support this study.

References

1 Thomée V., Galerkin Finite Element Methods for Parabolic Problems, 1997, Berlin, Germany, Springer.
10.1007/978-3-662-03359-3
Web of Science® Google Scholar
2 Wu J., Theory and Applications of Partial Functional-Differential Equations, 1996, Springer, New York, NY, USA.
10.1007/978-1-4612-4050-1
Google Scholar
3 Cannon J. R. and Lin Y., Non-classical H1 projection and Galerkin methods for non-linear parabolic integro-differential equations, Calcolo. (1988) 25, 187–201.
10.1007/BF02575943
Google Scholar
4 Li D. and Wang J., Unconditionally optimal error analysis of crank-nicolson galerkin fems for a strongly nonlinear parabolic system, Journal of Scientific Computing. (2017) 72, no. 2, 892–915, https://doi.org/10.1007/s10915-017-0381-3, MR3673699.
10.1007/s10915-017-0381-3
Web of Science® Google Scholar
5 Li B. and Sun W., Error analysis of linearized semi-implicit galerkin finite element methods for nonlinear parabolic equations, International Journal of Numerical Analysis & Modeling. (2013) 10, no. 3, 622–633, 2-s2.0-84878791453.
Web of Science® Google Scholar
6 Ascher U. M., Ruuth S. J., and Wetton B. T., Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis. (1995) 32, no. 3, 797–823, https://doi.org/10.1137/0732037, MR1335656, Zbl0841.65081.
10.1137/0732037
Web of Science® Google Scholar
7 Akrivis G. and Li B., Maximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolic equations, SIAM Journal on Numerical Analysis. (2017) https://doi.org/10.1093/imanum/drx008.
10.1093/imanum/drx008
PubMed Web of Science® Google Scholar
8 Higueras I., Strong stability for additive Runge-Kutta methods, SIAM Journal on Numerical Analysis. (2006) 44, no. 4, 1735–1758, 2-s2.0-34547535337, https://doi.org/10.1137/040612968.
10.1137/040612968
Web of Science® Google Scholar
9 Araujo A., A note on B-stability of splitting methods, Computing and Visualization in Science. (2004) 26, no. 2-3, 53–57, https://doi.org/10.1007/s00791-003-0108-x, MR2061265.
10.1007/s00791-003-0108-x
Google Scholar
10 Kennedy C. A. and Carpenter M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Applied Numerical Mathematics. (2003) 44, no. 1-2, 139–181, https://doi.org/10.1016/S0168-9274(02)00138-1, MR1951292, Zbl1013.65103.
10.1016/S0168-9274(02)00138-1
Web of Science® Google Scholar
11 Koto T., Stability of IMEX Runge-Kutta methods for delay differential equations, Journal of Computational and Applied Mathematics. (2008) 211, 201–212.
10.1016/j.cam.2006.11.011
Web of Science® Google Scholar
12 Liu H. and Zou J., Some new additive Runge-Kutta methods and their applications, Journal of Computational and Applied Mathematics. (2006) 190, no. 1-2, 74–98, https://doi.org/10.1016/j.cam.2005.02.020, MR2209495, Zbl1089.65066, 2-s2.0-31944447740.
10.1016/j.cam.2005.02.020
Web of Science® Google Scholar
13 Li D., Zhang C., and Ran M., A linear finite difference scheme for generalized time fractional Burgers equation, Applied Mathematical Modelling. (2016) 40, no. 11-12, 6069–6081.
10.1016/j.apm.2016.01.043
Web of Science® Google Scholar
14 Li D., Wang J., and Zhang J., Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM Journal on Scientific Computing. (2017) 39, no. 6, A3067–A3088, https://doi.org/10.1137/16M1105700, MR3738320.
10.1137/16M1105700
Web of Science® Google Scholar
15 Torelli L., Stability of numerical methods for delay differential equations, Journal of Computational and Applied Mathematics. (1989) 25, no. 1, 15–26, https://doi.org/10.1016/0377-0427(89)90071-X, MR977881, Zbl0664.65073, 2-s2.0-38249024166.
10.1016/0377-0427(89)90071-X
Web of Science® Google Scholar
16 in′t Hout K. J., Stability analysis of Runge-Kutta methods for systems of delay differential equations, IMA Journal of Numerical Analysis. (1997) 17, no. 1, 17–27, https://doi.org/10.1093/imanum/17.1.17, MR1427613.
10.1093/imanum/17.1.17
Web of Science® Google Scholar
17 Baker C. T. and Tang A., Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays, Applied Numerical Mathematics. (1997) 24, no. 2-3, 153–173, https://doi.org/10.1016/S0168-9274(97)00018-4, MR1464721, Zbl0878.65122.
10.1016/S0168-9274(97)00018-4
Web of Science® Google Scholar
18 Zhang C. and Vandewalle S., General linear methods for Volterra integro-differential equations with memory, SIAM Journal on Scientific Computing. (2006) 27, no. 6, 2010–2031, https://doi.org/10.1137/040607058, MR2211437, Zbl1104.65133, 2-s2.0-33751226927.
10.1137/040607058
Web of Science® Google Scholar
19 Li D. and Zhang C., Nonlinear stability of discontinuous Galerkin methods for delay differential equations, Applied Mathematics Letters. (2010) 23, no. 4, 457–461, https://doi.org/10.1016/j.aml.2009.12.003, MR2594863, Zbl1191.65111, 2-s2.0-76749166248.
10.1016/j.aml.2009.12.003
Web of Science® Google Scholar
20 Li D. and Zhang C., L∞ error estimates of discontinuous Galerkin methods for delay differential equations, Applied Numerical Mathematics. (2014) 82, 1–10, https://doi.org/10.1016/j.apnum.2014.01.008, MR3212375.
10.1016/j.apnum.2014.01.008
Web of Science® Google Scholar
21 Barwell V. K., Special stability problems for functional differential equations, BIT. (1975) 15, 130–135.
10.1007/BF01932685
Google Scholar
22 Bellen A. and Zennaro M., Strong contractivity properties of numerical methods for ordinary and delay differential equations, Applied Numerical Mathematics. (1992) 9, no. 3-5, 321–346, https://doi.org/10.1016/0168-9274(92)90025-9, MR1158492, Zbl0749.65042.
10.1016/0168-9274(92)90025-9
Web of Science® Google Scholar
23 Burrage K., High order algebraically stable Runge-Kutta methods, BIT. (1978) 18, no. 4, 373–383, https://doi.org/10.1007/BF01932017, MR520749.
10.1007/BF01932017
Google Scholar
24 Burrage K. and Butcher J. C., Nonlinear stability of a general class of differential equation methods, BIT. (1980) 20, no. 2, 185–203, https://doi.org/10.1007/BF01933191, MR583033.
10.1007/BF01933191
Google Scholar
25 Cooper G. J. and Sayfy A., Additive Runge-Kutta methods for stiff ordinary differential equations, Mathematics of Computation. (1983) 40, no. 161, 207–218, https://doi.org/10.2307/2007370, MR679441, Zbl0525.65053, 2-s2.0-84966229401.
10.1090/S0025-5718-1983-0679441-1
Web of Science® Google Scholar
26 Dekker K. and Verwer J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, 1984, North-Holland Publishing, Amsterdam, The Netherlands, MR774402, Zbl0571.65057.
Google Scholar
27 Ferracina L. and Spijker M. N., Strong stability of singly-diagonally-implicit Runge-Kutta methods, Applied Numerical Mathematics. (2008) 58, no. 11, 1675–1686, https://doi.org/10.1016/j.apnum.2007.10.004, MR2458475, Zbl1153.65080.
10.1016/j.apnum.2007.10.004
Web of Science® Google Scholar
28 in′t Hout K. J. and Spijker M. N., K. Strehmel, The θ-methods in the numerical solution of delay differential equations, The Numerical Treatment of Differential Equations, 1991, 121, 61–67.
Google Scholar
29 Zennaro M., Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations, Numerische Mathematik. (1997) 77, no. 4, 549–563, https://doi.org/10.1007/s002110050300, MR1473396, Zbl0886.65092, 2-s2.0-0031501318.
10.1007/s002110050300
Web of Science® Google Scholar
30 Li D., Zhang C., and Wang W., Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Analysis: Real World Applications. (2012) 13, no. 3, 1401–1415, https://doi.org/10.1016/j.nonrwa.2011.11.005, MR2863967, Zbl1239.35069.
10.1016/j.nonrwa.2011.11.005
Web of Science® Google Scholar
31 Garcia-Celayeta B., Higueras I., and Roldan T., Contractivity/monotonicity for additive Range-kutta methods: Inner product norms, Applied Numerical Mathematics. (2006) 56, no. 6, 862–878, https://doi.org/10.1016/j.apnum.2005.07.001, MR2222124.
10.1016/j.apnum.2005.07.001
Web of Science® Google Scholar
32 Huang C., Dissipativity of one-leg methods for dynamical systems with delays, Applied Numerical Mathematics. (2000) 35, no. 1, 11–22, https://doi.org/10.1016/S0168-9274(99)00048-3, MR1784067, Zbl0965.65136.
10.1016/S0168-9274(99)00048-3
Web of Science® Google Scholar
33 Zhang C. and Zhou S., Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations, Journal of Computational and Applied Mathematics. (1997) 85, no. 2, 225–237, 2-s2.0-0031276989, https://doi.org/10.1016/S0377-0427(97)00118-0, Zbl0904.65082.
10.1016/S0377-0427(97)00118-0
Web of Science® Google Scholar
34 Huang C., Li S., Fu H., and Chen G., Nonlinear stability of general linear methods for delay differential equations, BIT Numerical Mathematics. (2002) 42, no. 2, 380–392, https://doi.org/10.1023/A:1021955126558, MR1912593, Zbl1009.65054.
10.1023/A:1021955126558
Web of Science® Google Scholar
35 Yuan H., Zhao J., and Xu Y., Nonlinear stability and D-convergence of additive Runge-Kutta methods for multidelay-integro-differential equations, Abstract and Applied Analysis. (2012) 2012, 22, 854517, https://doi.org/10.1155/2012/854517.
10.1155/2012/854517
Google Scholar

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Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

Abstract

1. Introduction

2. The Numerical Methods

3. Stability Analysis

4. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

Abstract

1. Introduction

2. The Numerical Methods

3. Stability Analysis

4. Conclusion

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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