A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes

Abstract

Runge-Kutta 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 problem-independent and scale-free but they can indiscriminately clip extrema, resulting in degraded accuracy. Those based on total variation bounded (TVB) limiters are neither problem-independent nor scale-free. In order to get past these limitations we realize that the solution in RKDG schemes can carry meaningful sub-structure within a zone that may not need to be limited. To make this sub-structure visible, we take a sub-cell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful sub-structure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scale-free and problem-independent. It, therefore, overcomes some of the limitations of previously-used strategies for detecting troubled zones. The moments of the troubled zones can then be corrected using a weighted essentially non-oscillatory (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, low-storage alternatives to RKDG schemes. Particular attention has been paid to the reconstruction of cross-terms in multi-dimensional 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 large-scale parallel adaptive mesh refinement calculations.

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

“…In the discontinuous Galerkin approach, on the other hand, even if in principle no spatial reconstruction is needed, in practice it is necessary to introduce some sort of limiters to avoid oscillations in the presence of discontinuities. Among the most relevant limiters proposed so far we mention the use of artificial viscosity [66,98,26,56,38,55], of spectral filtering [103], of (H) WENO limiting procedures [102,101,77,4,78,79,68], and of slope and moment limiting [30,88,104,1,125,37]. In [53] we have recently proposed a totally different and alternative solution to this longstanding problem, which relies on a new a posteriori sub-cell finite volume limiting approach.…”

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

“…From one side, it is possible to introduce additional numerical dissipation, either in the form of artificial viscosity (R. Hartmann & P.Houston 2002;Persson & Peraire 2006;Cesenek et al 2013), or by means of filtering (Radice & Rezzolla 2011). From another side, it is possible to isolate the so-called troubled cells, namely those affected by spurious oscillations, and adopt for them some sort of nonlinear finite-volume-type slope-limiting procedure (Cockburn & Shu 1998;J.Qiu & C-W.Shu 2004;Balsara et al 2007; J. Zhu et al 2008;J.Zhu & Qiu 2013;H.Luo et al 2007; L. Krivodonova 2007), either based on nonlinear WENO/HWENO reconstruction or by applying a TVB limiter to the higher order moments of the discrete solution.…”

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

“…Employing a first-order backward Euler implicit time stepping scheme, equation (27)t a k e st h ef o l l o w i n gf o r m…”

“…Furthermore, several research groups extended the original schemes to handle unstructured [18][19][20][21][22]a sw e l la sm i x e d -e l e m e n tu n s t r u c t u r e dg r i d s [ 3,[23][24][25][26], furthermore WENO methods are also utilised within the DG framework [27,28].…”