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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
ODEPACK
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##### 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)
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