×

zbMATH — the first resource for mathematics

On the local and global errors of splitting approximations of reaction-diffusion equations with high spatial gradients. (English) Zbl 1122.65061
The authors study the approximation by splitting techniques of the ordinary differential equation \(\dot{U} +AU+BU=0\), \(U(0)=U_0\) with \(A\) and \(B\) two matrices. They consider a stiff problem in the sense that \(A\) is ill-conditionned and \(U_0\) is a vector which is the discretization of a function with a very high derivative. This situation may appear for example when we study the discretization of a partial differential equation. Some error estimates for two general matrices are obtained, especially in the stiff case, where the estimates are independent of \(U_0\) and the commutator between \(A\) and \(B\).

MSC:
65L05 Numerical methods for initial value problems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
34A30 Linear ordinary differential equations and systems, general
35K57 Reaction-diffusion equations
Software:
ODEPACK
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/BF00281235 · Zbl 0113.32303 · doi:10.1007/BF00281235
[2] DOI: 10.1137/0705041 · Zbl 0184.38503 · doi:10.1137/0705041
[3] Hairer E., Geometric Numerical Integration 31, 2. ed. (2006)
[4] DOI: 10.1023/A:1022396519656 · Zbl 0972.65061 · doi:10.1023/A:1022396519656
[5] DOI: 10.1016/S0021-7824(01)01216-8 · Zbl 1030.35095 · doi:10.1016/S0021-7824(01)01216-8
[6] DOI: 10.1016/j.laa.2003.09.010 · Zbl 1054.17005 · doi:10.1016/j.laa.2003.09.010
[7] DOI: 10.1090/S0025-5718-00-01277-1 · Zbl 0981.65107 · doi:10.1090/S0025-5718-00-01277-1
[8] DOI: 10.1007/s002200100376 · Zbl 0996.47046 · doi:10.1007/s002200100376
[9] DOI: 10.1007/s002200100376 · Zbl 0996.47046 · doi:10.1007/s002200100376
[10] Kolmogoroff A., Bulletin de l’Université d’état à Moscou, Série Internationale Section A Mathématiques et Mécanique 1 pp 1– (1937)
[11] Gray P., Chemical Oscillations and Instabilites (1994)
[12] Volpert A. I., Traveling Wave Solutions of Parabolic Systems (1994) · Zbl 1017.34014
[13] Hindmarsh A., ODEPACK, a Systematized Collection of ODE Solvers, in Scientific Computing (1983)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.