TY - JOUR
T1 - On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations
AU - Debrabant, Kristian
AU - Lang, Jens
PY - 2015
Y1 - 2015
N2 - The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer[SIAM J. Sci. Comput. 29, 2007] 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.
AB - The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer[SIAM J. Sci. Comput. 29, 2007] 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.
KW - Asymptotic global error control
KW - Asymptotic global error estimation
KW - Defects and local errors
KW - Finite difference method
KW - Method of lines
KW - Numerical integration for PDEs
U2 - 10.1016/j.cma.2014.11.032
DO - 10.1016/j.cma.2014.11.032
M3 - Journal article
SN - 0045-7825
VL - 288
SP - 110
EP - 126
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -