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On asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations. (English) Zbl 1425.65087
Summary: The aim of this paper is to extend the global error estimation and control addressed in [J. Lang and J. G. Verwer, SIAM J. Sci. Comput. 29, No. 4, 1460–1475 (2007; Zbl 1145.65047)] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete \(L_2\)-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.
MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
ROS3P
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References:
[1] Lang, J.; Verwer, J. G., On global error estimation and control for initial value problems, SIAM J. Sci. Comput., 29, 1460-1475, (2007) · Zbl 1145.65047
[2] Skeel, R. D., Thirteen ways to estimate global error, Numer. Math., 48, 1-20, (1986) · Zbl 0562.65050
[3] Wouver, A. Vande; Saucez, P.; Schiesser, W. E., Some user-oriented comparisons of adaptive grid methods for partial differential equations in one space dimension, Appl. Numer. Math., 26, 49-62, (1998) · Zbl 0890.65100
[4] Wouver, A. Vande; Saucez, P.; Schiesser, W. E.; Thompson, S., A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines, J. Comput. Appl. Math., 183, 245-258, (2005) · Zbl 1071.65544
[5] Lang, J., (Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm and Applications, Lecture Notes in Computational Science and Engineering, vol. 16, (2000), Springer)
[6] Nowak, U., A fully adaptive MOL-treatment of parabolic 1D-problems with extrapolation techniques, Appl. Numer. Math., 20, 129-145, (1996) · Zbl 0939.65108
[7] Schönauer, W.; Schnepf, E.; Raith, K., Experiences in designing PDE software with selfadaptive variable step size/order difference methods, Computing, 5, 227-242, (1984) · Zbl 0565.65073
[8] Lawson, L.; Berzins, M.; Dew, P. M., Balancing space and time errors in the method of lines for parabolic equations, SIAM J. Sci. Stat. Comput., 12, 573-594, (1991) · Zbl 0725.65087
[9] Schmich, M.; Vexler, B., Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Stat. Comput., 30, 369-393, (2008) · Zbl 1169.65098
[10] Larsson, S.; Thomée, V., (Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics, vol. 45, (2005), Springer)
[11] Thomas, J. W., (Numerical Partial Differential Equations. Finite Difference Methods, Texts in Applied Mathematics, vol. 22, (1995), Springer) · Zbl 0831.65087
[12] Berzins, M., Global error estimation in the method of lines for parabolic equations, SIAM J. Sci. Stat. Comput., 9, 687-703, (1988) · Zbl 0659.65081
[13] K. Debrabant, J. Lang, On global error estimation and control of finite difference solutions for parabolic equations, in: Adaptive Modeling and Simulation 2013—Proceedings of the 6th International Conference on Adaptive Modeling and Simulation, ADMOS 2013, 2013, pp. 187-198.
[14] Shampine, L. F., Numerical solution of ordinary differential equations, (1994), Chapman & Hall New York · Zbl 0832.65063
[15] Lang, J.; Verwer, J. G., ROS3P—an accurate third-order rosenbrock solver designed for parabolic problems, BIT, 41, 731-738, (2001) · Zbl 0996.65099
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