A Moving Mesh WENO Method for One-Dimensional Conservation Laws

Abstract

Abstract. In this paper, we develop an efficient moving mesh weighted essentially nonoscillatory (WENO) method for one-dimensional hyperbolic conservation laws. The method is based on the quasi-Lagrange approach of the moving mesh strategy in which the mesh is considered to move continuously in time. Several issues arising from the implementation of the scheme, including mesh smoothness, mesh movement restriction, and computation of transformation relations, and their effects on the accuracy of the underlying scheme have been addressed. Particularly, it is found that a least squares smoothing can be used to effectively smooth the mesh, and the transformation relations can be computed using either high order finite differences or WENO applied to some geometric conservation laws. Moreover, mesh movement can cause WENO schemes to become unconditionally unstable. A simple strategy is used to restrict the mesh movement and recover the stability. Numerical results are presented to demonstrate the accuracy and shock-capturing ability of the new scheme.

An Eulerian–Lagrangian Weighted Essentially Nonoscillatory scheme for nonlinear conservation laws

“…We remark that the moving mesh methods (see, e.g., [26,27]) are similar to Eulerian-Lagrangian methods, and the adaptive WENO methods [28] are related to solution re-averaging, and so share many similar ideas to what we present herein.…”

A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws

“…Moreover, it is shown that the numerical solution with a moving mesh is generally more accurate than that with a uniform mesh of the same number of points and often comparable with the solution obtained with a much finer uniform mesh. Furthermore, numerical results for problems with smooth and discontinuous solutions in one and two dimensions have shown that the accuracy of the method is not sensitive to the smoothness of the mesh, which is in contrast with the situation for the moving mesh finite difference WENO method [38].…”

“…It is common practice in moving mesh computation to smooth the metric tensor/monitor function for smoother meshes. To this end, we apply a low-pass filter [17,38] to the smoothing of the metric tensor several sweeps every time it is computed.…”

“…Moving and uniform meshes will be denoted by "MM" and "UM", respectively. Unless otherwise stated, three sweeps of a low-pass filter [17,38] are applied to the smoothing of the metric tensor every time it is computed. In addition, for examples having an exact solution, the error of the computed solution is measured in the global norm, i.e., subject to the initial condition u(x, 0) = 0.5+sin(πx) and a periodic boundary condition.…”

“…Moreover, the newly developed discretization of the MMPDE [13] is used here, which makes the implementation more easier and much more reliable since there is a theoretical guarantee for mesh nonsingularity [14]. In contrast to the moving mesh finite difference WENO method [38] where the smoothness of the mesh is extremely important for accuracy, we find that the moving mesh DG method presented in this work is not sensitive to the smoothness of the mesh. Furthermore, the method discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one.…”