High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

Year:    2021

Author:    Jun Zhu, Chi-Wang Shu, Jianxian Qiu

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 1030–1058

Abstract

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0096

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 1030–1058

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    Runge-Kutta discontinuous Galerkin method multi-resolution WENO limiter tetrahedral mesh.

Author Details

Jun Zhu

Chi-Wang Shu

Jianxian Qiu

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