A subcell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes
Abstract
RungeKutta Discontinuous Galerkin (RKDG) schemes can provide highly accurate solutions for a large class of important scientific problems. Using them for problems with shocks and other discontinuities requires that one has a strategy for detecting the presence of these discontinuities. Strategies that are based on total variation diminishing (TVD) limiters can be problemindependent and scalefree but they can indiscriminately clip extrema, resulting in degraded accuracy. Those based on total variation bounded (TVB) limiters are neither problemindependent nor scalefree. In order to get past these limitations we realize that the solution in RKDG schemes can carry meaningful substructure within a zone that may not need to be limited. To make this substructure visible, we take a subcell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful substructure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scalefree and problemindependent. It, therefore, overcomes some of the limitations of previouslyused strategies for detecting troubled zones. The moments of the troubled zones can then be corrected using a weighted essentially nonoscillatory (WENO) or Hermite WENO (HWENO) approach. In the course of doing this work it was also realized that the most significant variation in the solution is contained in the solution variables and their first moments. Thus the additional moments can be reconstructed using the variables and their first moments, resulting in a very substantial savings in computer memory. We call such schemes hybrid RKDG+HWENO schemes. It is shown that such schemes can attain the same formal accuracy as RKDG schemes, making them attractive, lowstorage alternatives to RKDG schemes. Particular attention has been paid to the reconstruction of crossterms in multidimensional problems and explicit, easy to implement formulae have been catalogued for third and fourth order of spatial accuracy. The utility of hybrid RKDG+WENO schemes has been illustrated with several stringent test problems in one and two dimensions. It is shown that their accuracy is usually competitive with the accuracy of RKDG schemes of the same order. Because of their compact stencils and low storage, hybrid RKDG+HWENO schemes could be very useful for largescale parallel adaptive mesh refinement calculations.
 Publication:

Journal of Computational Physics
 Pub Date:
 September 2007
 DOI:
 10.1016/j.jcp.2007.04.032
 Bibcode:
 2007JCoPh.226..586B