Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws

Abstract

Abstract. We develop efficient moving mesh algorithms for one-and two-dimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and mesh-redistribution. The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula proposed in this work. The iteration for the meshredistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.Key words. adaptive mesh method, hyperbolic conservation laws, finite volume method AMS subject classifications. 65M93, 35L64, 76N10PII. S003614290138437X 1. Introduction. Adaptive mesh methods have important applications for a variety of physical and engineering areas such as solid and fluid dynamics, combustion, heat transfer, material science, etc. The physical phenomena in these areas develop dynamically singular or nearly singular solutions in fairly localized regions, such as shock waves, boundary layers, detonation waves, etc. The numerical investigation of these physical problems may require extremely fine meshes over a small portion of the physical domain to resolve the large solution variations. In multidimensions, developing effective and robust adaptive grid methods for these problems becomes necessary. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost. In the past two decades, there has been important progress in developing mesh methods for PDEs, including the variational approach of Winslow [36] Mackenzie, and Russell [33]. However, many existing moving mesh methods for hyperbolic problems are designed for one space dimension. In one dimension, it is generally possible to compute on a very fine grid, and so the need for moving mesh methods may not be clear. Multidimensional moving mesh methods are often difficult to use in fluid dynamics problems, since the grid will typically suffer large distortions and possible tangling. It is therefore useful to design a simple a...

An alternating Crank–Nicolson method for the numerical solution of the phase‐field equations using adaptive moving meshes

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