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Volume 33, Issue 5
High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

Yizhou Lu, Jun Zhu, Shengzhu Cui, Zhenming Wang, Linlin Tian & Ning Zhao

Commun. Comput. Phys., 33 (2023), pp. 1217-1239.

Published online: 2023-06

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  • Abstract

In this paper, a new multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter uses information of the LDG solution essentially only within the troubled cell itself, to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

  • AMS Subject Headings

65M60, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-1217, author = {Lu , YizhouZhu , JunCui , ShengzhuWang , ZhenmingTian , Linlin and Zhao , Ning}, title = {High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {5}, pages = {1217--1239}, abstract = {

In this paper, a new multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter uses information of the LDG solution essentially only within the troubled cell itself, to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

}, issn = {1991-7120}, doi = {https://doi.org/ 10.4208/cicp.OA-2022-0096}, url = {http://global-sci.org/intro/article_detail/cicp/21760.html} }
TY - JOUR T1 - High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes AU - Lu , Yizhou AU - Zhu , Jun AU - Cui , Shengzhu AU - Wang , Zhenming AU - Tian , Linlin AU - Zhao , Ning JO - Communications in Computational Physics VL - 5 SP - 1217 EP - 1239 PY - 2023 DA - 2023/06 SN - 33 DO - http://doi.org/ 10.4208/cicp.OA-2022-0096 UR - https://global-sci.org/intro/article_detail/cicp/21760.html KW - Local discontinuous Galerkin method, multi-resolution WENO limiter, triangular meshes, Navier-Stokes equations. AB -

In this paper, a new multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter uses information of the LDG solution essentially only within the troubled cell itself, to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

Yizhou Lu, Jun Zhu, Shengzhu Cui, Zhenming Wang, Linlin Tian & Ning Zhao. (2023). High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes. Communications in Computational Physics. 33 (5). 1217-1239. doi: 10.4208/cicp.OA-2022-0096
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