On the stability of implicit-explicit linear multistep methods. (English) Zbl 0887.65094
This paper deals with the numerical solution of large systems of ordinary differential equations with both stiff and nonstiff parts using implicit-explicit linear multistep methods, i.e., combinations of two methods. It is assumed that the individual implicit method is \(A\)-stable. Stability properties of the implicit-explicit schemes for the scalar test equation \(w'(t)= \lambda w(t)+ \mu w(t)\) are investigated, where \(\lambda\) and \(\mu\) represent the eigenvalues of the nonstiff and stiff part, respectively.
The following questions are discussed: stability under the assumption that \(\lambda\) lies in the stability region of the individual explicit method, and restrictions on \(\lambda\) for having stability for arbitrary \(\mu\) in the left half-plane. Several popular second-order methods are studied.
The following questions are discussed: stability under the assumption that \(\lambda\) lies in the stability region of the individual explicit method, and restrictions on \(\lambda\) for having stability for arbitrary \(\mu\) in the left half-plane. Several popular second-order methods are studied.
Reviewer: R.Scherer (Karlsruhe)
MSC:
65L20 | Stability and convergence of numerical methods for ordinary differential equations |
65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
34A34 | Nonlinear ordinary differential equations and systems |
34E13 | Multiple scale methods for ordinary differential equations |
Keywords:
implicit-explicit methods; linear multistep methods; methods of lines; systems with stiff and nonstiff parts; linear test equation; stabilitySoftware:
RODASReferences:
[1] | Akrivis, G.; Crouzeix, M.; Makridadis, C., Implicit-explicit multistep finite element methods for nonlinear parabolic equations, (Report 95-22 (1995), University of Rennes) |
[2] | Ascher, U. M.; Ruuth, S. J.; Wetton, B., Implicit-explicit methods for time-dependent PDE’s, SIAM J. Numer. Anal., 32, 797-823 (1995) · Zbl 0841.65081 |
[3] | Crouzeix, M., Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques, Numer. Math., 35, 257-276 (1980) · Zbl 0419.65057 |
[4] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems (1987), Springer: Springer Berlin · Zbl 0638.65058 |
[5] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1991), Springer: Springer Berlin · Zbl 0729.65051 |
[6] | Hundsdorfer, W.; Verwer, J. G., A note on splitting errors for advection-reaction equations, Appl. Numer. Math., 18, 191-199 (1995) · Zbl 0833.65099 |
[7] | Lambert, J. D., Numerical Methods for Ordinary Differential Systems (1991), Wiley: Wiley Chichester · Zbl 0745.65049 |
[8] | Nevanlinna, O.; Liniger, W., Contractive methods for stiff differential equations II, BIT, 19, 53-72 (1979) · Zbl 0408.65046 |
[9] | Varah, J. M., Stability restrictions on second order, three-level finite-difference schemes for parabolic equations, SIAM J. Numer. Anal., 17, 300-309 (1980) · Zbl 0426.65048 |
[10] | Verwer, J. G.; Blom, J. G.; Hundsdorfer, W., An implicit-explicit approach for atmospheric transport-chemistry problems, Appl. Numer. Math., 20, 191-209 (1996) · Zbl 0853.76092 |
[11] | Wesseling, P., Von Neumann stability conditions for the convection-diffusion equation, IMA J. Numer. Anal., 16, 583-598 (1996) · Zbl 0862.65046 |
[12] | Zlatev, Z., Computer Treatment of Large Air Pollution Models (1995), Kluwer: Kluwer Dordrecht · Zbl 0852.65058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.