Volume 5, Issue 3
A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods

Adv. Appl. Math. Mech., 5 (2013), pp. 365-390.

Published online: 2013-05

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• Abstract

Discontinuities usually appear in solutions of nonlinear conservation laws even though the initial condition is smooth, which leads to great difficulty in computing these solutions numerically. The Runge-Kutta discontinuous Galerkin (RKDG) methods are efficient methods for solving nonlinear conservation laws, which are high-order accurate and highly parallelizable, and can be easily used to handle complicated geometries and boundary conditions. An important component of RKDG methods for solving nonlinear conservation laws with strong discontinuities in the solution is a nonlinear limiter, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods. A limiter contains two parts, first to identify the "troubled cells", namely, those cells which might need the limiting procedure, then to replace the solution polynomials in those troubled cells by reconstructed polynomials which maintain the original cell averages (conservation). [SIAM J. Sci. Comput., 26 (2005), pp. 995-1013] focused on discussing the first part of limiters. In this paper, focused on the second part, we will systematically investigate and compare a few different reconstruction strategies with an objective of obtaining the most efficient and reliable reconstruction strategy. This work can help with the choosing of right limiters so one can resolve sharper discontinuities, get better numerical solutions and save the computational cost.

65M60, 65M99, 35L65

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@Article{AAMM-5-365, author = {Zhu , HongqiangCheng , Yue and Qiu , Jianxian}, title = {A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {3}, pages = {365--390}, abstract = {

Discontinuities usually appear in solutions of nonlinear conservation laws even though the initial condition is smooth, which leads to great difficulty in computing these solutions numerically. The Runge-Kutta discontinuous Galerkin (RKDG) methods are efficient methods for solving nonlinear conservation laws, which are high-order accurate and highly parallelizable, and can be easily used to handle complicated geometries and boundary conditions. An important component of RKDG methods for solving nonlinear conservation laws with strong discontinuities in the solution is a nonlinear limiter, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods. A limiter contains two parts, first to identify the "troubled cells", namely, those cells which might need the limiting procedure, then to replace the solution polynomials in those troubled cells by reconstructed polynomials which maintain the original cell averages (conservation). [SIAM J. Sci. Comput., 26 (2005), pp. 995-1013] focused on discussing the first part of limiters. In this paper, focused on the second part, we will systematically investigate and compare a few different reconstruction strategies with an objective of obtaining the most efficient and reliable reconstruction strategy. This work can help with the choosing of right limiters so one can resolve sharper discontinuities, get better numerical solutions and save the computational cost.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2012.m22}, url = {http://global-sci.org/intro/article_detail/aamm/75.html} }
TY - JOUR T1 - A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods AU - Zhu , Hongqiang AU - Cheng , Yue AU - Qiu , Jianxian JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 365 EP - 390 PY - 2013 DA - 2013/05 SN - 5 DO - http://doi.org/10.4208/aamm.2012.m22 UR - https://global-sci.org/intro/article_detail/aamm/75.html KW - Limiter, discontinuous Galerkin method, hyperbolic conservation laws. AB -

Discontinuities usually appear in solutions of nonlinear conservation laws even though the initial condition is smooth, which leads to great difficulty in computing these solutions numerically. The Runge-Kutta discontinuous Galerkin (RKDG) methods are efficient methods for solving nonlinear conservation laws, which are high-order accurate and highly parallelizable, and can be easily used to handle complicated geometries and boundary conditions. An important component of RKDG methods for solving nonlinear conservation laws with strong discontinuities in the solution is a nonlinear limiter, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods. A limiter contains two parts, first to identify the "troubled cells", namely, those cells which might need the limiting procedure, then to replace the solution polynomials in those troubled cells by reconstructed polynomials which maintain the original cell averages (conservation). [SIAM J. Sci. Comput., 26 (2005), pp. 995-1013] focused on discussing the first part of limiters. In this paper, focused on the second part, we will systematically investigate and compare a few different reconstruction strategies with an objective of obtaining the most efficient and reliable reconstruction strategy. This work can help with the choosing of right limiters so one can resolve sharper discontinuities, get better numerical solutions and save the computational cost.

Hongqiang Zhu, Yue Cheng & Jianxian Qiu. (1970). A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods. Advances in Applied Mathematics and Mechanics. 5 (3). 365-390. doi:10.4208/aamm.2012.m22
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