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.

[1]  J. Remacle,et al.  Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws , 2004 .

[2]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[3]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[4]  Pierre Sagaut,et al.  A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods , 2001 .

[5]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[6]  Huanan Yang,et al.  An artificial compression method for ENO schemes - The slope modification method. [essentially nonoscillatory , 1990 .

[7]  Chi-Wang Shu,et al.  A technique of treating negative weights in WENO schemes , 2000 .

[8]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[9]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[10]  J. Flaherty,et al.  Parallel, adaptive finite element methods for conservation laws , 1994 .

[11]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[12]  Arne Taube,et al.  Arbitrary High-Order Discontinuous Galerkin Schemes for the Magnetohydrodynamic Equations , 2007, J. Sci. Comput..

[13]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[14]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case , 2005 .

[15]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[16]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[17]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[18]  Manuel Torrilhon,et al.  High order WENO schemes: investigations on non-uniform converges for MHD Riemann problems , 2004 .

[19]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[20]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[21]  Michael Dumbser,et al.  Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes , 2006, J. Sci. Comput..

[22]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[23]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[24]  Takashi Yabe,et al.  3. Cubic Interpolated Pseudo-Particle Method (CIP) for Solving Hyperbolic-Type Equation (II. Basic Algorithm) , 1985 .

[25]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[26]  A. Stroud,et al.  Gaussian quadrature formulas , 1966 .

[27]  Dinshaw S. Balsara,et al.  Highly parallel structured adaptive mesh refinement using parallel language-based approaches , 2001, Parallel Comput..

[28]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[29]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[30]  Dinshaw Balsara,et al.  Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction , 2003, astro-ph/0308249.

[31]  Takashi Yabe,et al.  Cubic interpolated pseudo-particle method (CIP) for solving hyperbolic-type equations , 1985 .

[32]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[33]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

[34]  M. Dumbser,et al.  Arbitrary high order discontinuous Galerkin schemes , 2005 .