On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations
It provides a method for estimating and controlling the overall error in numerical solutions of semilinear parabolic PDEs, which is useful for computational scientists requiring reliable error bounds.
This paper extends global error estimation and control from initial value problems to finite difference solutions of semilinear parabolic PDEs, combining temporal error estimation with spatial truncation error estimation via Richardson extrapolation. Numerical examples demonstrate the reliability of the proposed strategies.
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.