A New MATLAB Implementation And Analysis of A Moving Grid Method For Systems of One-Dimensional Time-Dependent Partial Differential Equations Based on The Equidistribution Principle

摘要

In the past several years, several numerical techniques have been developed to solve time-dependent partial differential equations (PDEs) in one dimension having solutions with steep gradients in space and in time. One of these techniques, a moving grid method, has been coupled with a spatial discretization method that automatically discretizes the spatial part of the user-defined PDE following the method of line approach. The objective of this paper is to report on the development of a Method of Lines (MOL) toolbox within MATLAB, and especially, on the implementation and test of a moving grid graphical user interface based on the equidistribution principle. This new user-interface includes various spatial approximation schemes based on finite differences and slope limiters, the choice between several monitor functions, automatic grid adaptation to the initial condition and provides a relatively easy tuning for the non-expert user. Several issues, including the sensitivity of the numerical results to the tuning parameters, are discussed. Some numerical examples characterized by solutions with steep moving fronts, are investigated so as to demonstrate the algorithm performance and to illustrate the simple and effective use of this software interface.