A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods

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.

A hybrid Hermite WENO scheme for hyperbolic conservation laws

“…Next, Li and Qiu [23] studied the hybrid WENO scheme using different switching principles [30], which illustrated that the free parameters troubled-cells indicator introduced by Krivodonova et al [19] (KXRCF) has the ability to identify the discontinuities well. Other different schemes introduced by [39,37,44,17] for hyperbolic conservation laws also showed the good performances of the KXRCF troubled-cell indicator. In this paper, we would choose it as the indictor to identify the troubled-cell where the solutions may be discontinuous.…”

Multi-layer Perceptron Estimator for the Total Variation Bounded Constant in Limiters for Discontinuous Galerkin Methods

“…Examples include the minmod-based total variation diminishing (TVD) limiters [14,25], the minmod-based total variation bounded (TVB) limiter [34], the moment limiter [1], the monotonicity-preserving limiter [38], and the weighted essentially non-oscillatory (WENO) limiter [28]. A summary and comparison of limiters can found in [44].…”

Automated parameters for troubled-cell indicators using outlier detection

“…This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically such that the indicator works well in a variety of situations [42]. Similarly, a parameter is required for adaptive mesh refinement [11].…”

“…The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of non-physical spurious oscillations. In general, many tests are required to obtain this optimal parameter for each problem [28,42].…”