A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods (Advances in Applied Mathematics and Mechanics, 2013)
Discontinuities usually appear insolutions of nonlinear conservation laws even though the initialcondition is smooth, which leads to great difficulty in computingthese solutions numerically. The Runge-Kutta discontinuous Galerkin(RKDG) methods are efficient methods for solving nonlinearconservation laws, which are high-order accurate and highlyparallelizable, and can be easily used to handle complicatedgeometries and boundary conditions. An important component of RKDGmethods for solving nonlinear conservation laws with strongdiscontinuities in the solution is a nonlinear limiter, which isapplied to detect discontinuities and control spurious oscillationsnear such discontinuities. Many such limiters have been used in theliterature on RKDG methods. A limiter contains two parts, first toidentify the "troubled cells", namely, those cells which mightneed the limiting procedure, then to replace the solutionpolynomials in those troubled cells by reconstructed polynomialswhich maintain the original cell averages (conservation). [SIAMJ. Sci. Comput., 26 (2005), pp. 995-1013] focused on discussing thefirst part of limiters. In this paper, focused on the second part,we will systematically investigate and compare a few differentreconstruction strategies with an objective of obtaining the mostefficient and reliable reconstruction strategy. This work can helpwith the choosing of right limiters so one can resolve sharperdiscontinuities, get better numerical solutions and save thecomputational cost.