Unconditionally stable difference methods for delay partial differential equations. (English) Zbl 1272.65066

The authors study finite difference approximations of parabolic partial differential equations with time-delay. They show that a variant of the classical second-order central difference approximation of the diffusion operator together with an approximation of the delay argument through linear interpolation and with the trapezoidal rule or with the second-order backward differentiation formula for the time derivative lead to a discretization which unconditionally preserves the delay dependent asymptotic stability of the linear test problem. Some numerical experiments illustrate the theoretical findings.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R10 Partial functional-differential equations
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[1] Baker C.T.H., Ford N.J.: Some applications of the boundary-locus method and the method of D-partitions. IMA J. Numer. Anal. 11, 143–158 (1991) · Zbl 0726.65152
[2] Bellen A., Zennaro M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003) · Zbl 1038.65058
[3] Bickart T.A.: P-stable and P[{\(\alpha\)},{\(\beta\)}]-stable integration interpolation methods in the solution of retarded differential- difference equations. BIT 22, 464–476 (1982) · Zbl 0531.65044
[4] Bocharov G.A., Marchuk G.I., Romanyukha A.A.: Numerical solution by LMMs of stiff delay differential systems modelling an immune response. Numer. Math. 73, 131–148 (1996) · Zbl 0859.65077
[5] Brunner H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge Monographs on Applied and Computational Mathematics, vol. 15. Cambridge University Press, Cambridge (2004) · Zbl 1059.65122
[6] Diekmann O., Van Gils S.A., Verduin Lunel S.M., Walther H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, Berlin (1995) · Zbl 0826.34002
[7] Guglielmi N.: On the asymptotic stability properties of Runge–Kutta methods for delay differential equations. Numer. Math. 77, 467–485 (1997) · Zbl 0885.65092
[8] Guglielmi N.: Delay dependent stability regions of {\(\Theta\)}-methods for delay differential equations. IMA J. Numer. Anal. 18, 399–418 (1998) · Zbl 0909.65050
[9] Guglielmi N., Hairer E.: Order stars and stability for delay differential equations. Numer. Math. 83, 371–383 (1999) · Zbl 0937.65079
[10] Guglielmi N., Hairer E.: Geometric proofs of numerical stability for delay equations. IMA J. Numer. Anal. 21, 439–450 (2001) · Zbl 0976.65077
[11] Higham D., Sarder T.: Existence and stability of fixed points for a discretised nonlinear reaction–diffusion equation with delay. Appl. Numer. Math. 18, 155–173 (1995) · Zbl 0834.65079
[12] Huang C.: Delay-dependent stability of high order Runge–Kutta methods. Numer. Math. 111, 377–387 (2009) · Zbl 1167.65045
[13] Huang C., Vandewalle S.: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J. Sci. Comput. 25, 1608–1632 (2004) · Zbl 1064.65078
[14] in’t Hout K.J.: A new interpolation procedure for adapting Runge–Kutta methods to delay differential equations.. BIT 32, 634–649 (1992) · Zbl 0765.65069
[15] Jaffer, S. K.: Delay-dependent numerical stability of delay differential equations and Kreiss resolvent condition. PhD thesis, Harbin Institute of Technology, Harbin (2001)
[16] Koto T.: A stability property of A-stable natural Runge–Kutta methods for systems of delay differential equations. BIT 34, 262–267 (1994) · Zbl 0805.65083
[17] Koto T.: Stability of {\(\theta\)}-methods for delay integro-differential equations. J. Comput. Appl. Math. 161, 393–404 (2003) · Zbl 1042.65108
[18] Maset S.: Stability of Runge–Kutta methods for linear delay differential equations. Numer. Math. 87, 355–371 (2000) · Zbl 0979.65069
[19] Qiu L., Yang B., Kuang J.: The NGP-stability of Runge–Kutta methods for systems of neutral delay differential equations. Numer. Math. 81, 451–459 (1999) · Zbl 0918.65061
[20] Thomas J.W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York (1995) · Zbl 0831.65087
[21] van der Houwen P.J., Sommeijer B.P., Baker C.T.H.: On the stability of predictor–corrector methods for parabolic equations with delay. IMA J. Numer. Anal. 6, 1–23 (1986) · Zbl 0623.65094
[22] Watanabe D.S., Roth M.G.: The stability of difference formulas for delay differential equations. SIAM J. Numer. Anal. 22, 132–145 (1985) · Zbl 0571.65075
[23] Wu J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) · Zbl 0870.35116
[24] Wu S., Gan S.: Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput. Math. Appl. 55, 2426–2443 (2008) · Zbl 1142.45306
[25] Zennaro M.: P-stability of Runge–Kutta methods for delay differential equations. Numer. Math. 49, 305–318 (1986) · Zbl 0598.65056
[26] Zubik-Kowal B.: Stability in the numerical solution of linear parabolic equations with a delay term. BIT 41, 191–206 (2001) · Zbl 0984.65091
[27] Zubik-Kowal B., Vandewalle S.: Waveform relaxation for functional-differential equations. SIAM J. Sci. Comput. 21, 207–226 (1999) · Zbl 0945.65107
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