دانلود رایگان مقاله برآورد تقریبی خطای جهانی و کنترل از راه حل تفاضل محدود برای معادلات سهموی

Abstract

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 L2-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.

8. Summary

We have developed an error control strategy for finite difference solutions of parabolic equations, involving both temporal and spatial discretization errors. The global time error strategy discussed in [1] appears to provide an excellent starting point for the development of such an algorithm. The classical ODE approach used there and the principle of tolerance proportionality are combined with an efficient estimation of the spatial error and mesh adaptation to control the overall global error. Two approaches have been presented to handle spatial mesh improvement: (i) globally uniform refinement and (ii) local refinement and coarsening based on an equidistribution principle. Inspired by [12], we have used Richardson extrapolation to approximate the spatial truncation error within the method of lines. Our control strategy aims at balancing the spatial and temporal discretization error in order to achieve an accuracy imposed by the user. The key ingredients are: (i) linearized error transport equations equipped with sufficiently accurate defects to approximate the global time error and global spatial error and (ii) uniform or adaptive mesh refinement and local error control in time based on tolerance proportionality to achieve global error control. For illustration of the performance and effectiveness of our approach, we have implemented second-order finite differences in one space dimension and the example integrator ROS3P [15]. On the basis of three different test problems we could observe that our approach is very reliable, both with respect to estimation and control. Needless to say that spatial mesh adaptation locally in time is more efficient for solutions having a strongly nonuniform nature in space, especially if it varies over time. This is clearly visible for the travelling wave solution of the Allen–Cahn problem. However, optimized uniform strategies might also be of interest if users would like to extend their own software packages not having the option of dynamic adaptive mesh refinement to global error control.