Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

Fu, Pei; Schnücke, Gero; Xia, Yinhua

doi:10.1090/mcom/3417

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes
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Math. Comp. 88 (2019), 2221-2255 Request permission

Abstract:

In Klingenberg, Schnücke, and Xia (Math. Comp. 86 (2017), 1203–1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semidiscrete method the $\mathrm {L}^2$-stability will be proven. Furthermore, an error estimate which provides the suboptimal ($k+\frac {1}{2}$) convergence with respect to the $\mathrm {L}^{\infty }\!\left (0,T;\mathrm {L}^{2}\!\left (\Omega \right )\right )$-norm will be presented when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree $k$. The two-dimensional fully-discrete explicit method will be combined with the bound-preserving limiter developed by Zhang, Xia, and Shu (in J. Sci. Comput. 50 (2012), 29–62). This limiter does not affect the high-order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter, the validity of a discrete maximum principle will be proven. The numerical stability, robustness, and accuracy of the method will be shown by a variety of two-dimensional computational experiments on moving triangular meshes.

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