Adaptive Moving Grid Method for One-Dimensional Systems of Partial Differential Equations and Its Numerical Solution

摘要

In this paper we examine a scheme for choosing a moving mesh based on minimizing the time rate of change of the solution in the moving coordinates. We show how to apply this method to systems where the time derivatives cannot be solved for explicitly, writing the moving mesh equations in an implicit form. We give a geometrical interpretation of the moving mesh equation which exposes some of its weaknesses, and suggest some modifications based on this interpretation which increase the efficiency of the scheme. Upon discretization of the original PDE system coupled with the moving mesh equations, a system of differential/algebraic equations (DAEs) is generated. We discuss the solution of this system using the code DASSL, and show how changing the error estimate for time step selection from the usual strategy can significantly reduce the computation time. We introduce a new DAE code, DASPK, which combines the time integration methods of DASSL with the preconditioned GMRES method for solving large sparse linear systems. Finally, we present the results of some numerical experiments on reaction-diffusion equations which illustrate how well the resulting methods work. 16 refs., 2 figs., 2 tabs. (ERA citation 14:019152)