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A moving mesh method for one-dimensional hyperbolic conservation laws

Stockie, John M. and MacKenzie, John A. and Russell, Robert D. (2001) A moving mesh method for one-dimensional hyperbolic conservation laws. SIAM Journal on Scientific Computing, 22 (5). pp. 1791-1813. ISSN 1064-8275

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Abstract

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.