Moving-Mesh Methods for One-Dimensional Hyperbolic Problems Using CLAWPACK

1. I N T R O D U C T I O N We study high-resolution finite-volume methods for the one-dimensional conservation law qt + f (q ) z = ~ (x ,q ) (1) on a moving grid, where the interface x~ between grid cells varies with time t~. Figure 1 shows a typical moving grid over one time step. We show how the wave-propagation algorithms developed in [1] and implemented in the CLAWPACK software [2] can be modified to handle moving grids. and consider applications to gas dynamics in a tube with moving-piston boundary conditions and also a moving interface between two gases. With a finite volume method, our approximation consists of cell averages 1 f~ x,~+l '~ q(x, t~) dx, The first author would like to express his thanks for the support by a NATO-CNR grant (No. 217.28/01) and the kind hospitality of the Department of Applied Mathematics of the Washington University. The second author has" been supported in part by DOE Grant DE-FG03-96ER25292 and NSF Grants DMS-9505021 and DMS-9626645. 0898-1221/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by A2~4S-TE X PII: S0898-1221 (02)00339-5

[1]  D. J. Evans,et al.  An implicit difference scheme for a moving boundary hyperbolic problem , 1993 .

[2]  John M. Stockie,et al.  A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws , 2000, SIAM J. Sci. Comput..

[3]  Keith Miller,et al.  Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension , 1998, SIAM J. Sci. Comput..

[4]  Wu,et al.  Shock-wave propagation in a sonoluminescing gas bubble. , 1993, Physical review letters.

[5]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[6]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[7]  Robert D. Russell,et al.  Moving Mesh Techniques Based upon Equidistribution, and Their Stability , 1992, SIAM J. Sci. Comput..

[8]  Patrick Jenny,et al.  Correction of Conservative Euler Solvers for Gas Mixtures , 1997 .

[9]  R. LeVeque.  Wave Propagation Algorithms for Multidimensional Hyperbolic Systems , 1997 .

[10]  Randall J. LeVeque,et al.  A wave propagation method for three-dimensional hyperbolic conservation laws , 2000 .

[11]  Smadar Karni,et al.  Hybrid Multifluid Algorithms , 1996, SIAM J. Sci. Comput..

[12]  Stanly Steinberg,et al.  Flux-corrected transport in a moving grid , 1994 .

[13]  L. Petzold,et al.  Moving Mesh Methods with Upwinding Schemes for Time-Dependent PDEs , 1997 .

[14]  J. Kevorkian,et al.  Multiple‐scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations , 1998 .

[15]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

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