Some user-oriented comparisons of adaptive grid methods for partial differential equations in one space dimension

Abstract

Although the use of adaptive grid methods in one spatial dimension is now basically well understood and research is mainly devoted to problems in higher spatial dimensions, only a few methods are now available in the form of software packages and it may still be difficult for the user to find a path through the maze of methods published in the literature. In this paper, we examine five adaptive grid codes based on the method of lines, i.e., two local refinement algorithms and three moving grid algorithms. We illustrate and compare the use of the several methods with the numerical solution of a number of test problems from science and engineering, e.g., a model of a single-step reaction with diffusion, Burgers' equation, a model of flame propagation, and soliton solutions of the cubic Schrödinger equation. From the user point of view, we address important questions concerning the accuracy, temporal performance, ease of use (tuning) and simplicity of implementation.