On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution

Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.

[1]  I. Babuska,et al.  Analysis of Optimal Finite Element Meshes in R1 , 1979 .

[2]  Keith Miller,et al.  Moving Finite Elements. I , 1981 .

[3]  J. M. Sanz-Serna,et al.  On simple moving grid methods for one-dimensional evolutionary partial differential equations , 1988 .

[4]  Joke Blom,et al.  An adaptive moving grid method for one-dimensional systems of partial differential equations , 1989 .

[5]  J. Hansen,et al.  Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial differential equations , 1991 .

[6]  Weizhang Huang,et al.  Moving Mesh Methods Based on Moving Mesh Partial Differential Equations , 1994 .

[7]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[8]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[9]  Weizhang Huang,et al.  A moving collocation method for solving time dependent partial differential equations , 1996 .

[10]  D. M. Sloan,et al.  Solution of Evolutionary Partial Differential Equations Using Adaptive Finite Differences with Pseudospectral Post-processing , 1997 .

[11]  Robert D. Russell,et al.  Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems , 1998, SIAM J. Sci. Comput..

[12]  Weizhang Huang,et al.  A high dimensional moving mesh strategy , 1998 .

[13]  A moving mesh method in multiblock domains with application to a combustion problem , 1999 .

[14]  George Beckett,et al.  Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem , 2000 .

[15]  George Beckett,et al.  Uniformly convergent high order finite element solutions of a singularly perturbed reaction-diffusion equation using mesh equidistribution , 2001 .

[16]  J. Mackenzie,et al.  On a uniformly accurate finite difference approximation of a singulary peturbed reaction-diffusion problem using grid equidistribution , 2001 .

[17]  A. Ramage,et al.  Computational solution of two-dimensional unsteady PDEs using moving mesh methods , 2002 .