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

*(English)*Zbl 1124.65072Summary: 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 is 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 is 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.

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 is 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 is 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.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

65Y05 | Parallel numerical computation |

##### Keywords:

higher order schemes; conservation laws; numerical examples; Runge-Kutta discontinuous Galerkin (RKDG) schemes; shocks; total variation diminishing limiters; total variation bounded limiters; monotonicity preserving strategy; weighted essentially non-oscillatory; parallel adaptive mesh refinement
PDF
BibTeX
Cite

\textit{D. S. Balsara} et al., J. Comput. Phys. 226, No. 1, 586--620 (2007; Zbl 1124.65072)

Full Text:
DOI

##### References:

[1] | M. Abromowitz, I.E. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, vol. 55, 1964. |

[2] | Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, Journal of computational physics, 160, 405-452, (2000) · Zbl 0961.65078 |

[3] | Balsara, D.S.; Norton, C., Highly parallel structured adaptive mesh refinement using parallel language-based approaches, Journal of parallel computation, 27, 37-70, (2001) · Zbl 0971.68017 |

[4] | Balsara, D.S., Second order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophysical journal supplement, 151, 149-184, (2004) |

[5] | Biswas, R.; Devine, K.D.; Flaherty, J., Parallel adaptive finite element methods for conservation laws, Applied numerical mathematics, 14, 255-283, (1994) · Zbl 0826.65084 |

[6] | Burbeau, A.; Sagaut, P.; Bruneau, C.H., A problem-independent limiter for high-order runge – kutta discontinuous Galerkin methods, Journal of computational physics, 169, 111-150, (2001) · Zbl 0979.65081 |

[7] | Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of computation, 52, 411-435, (1989) · Zbl 0662.65083 |

[8] | Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, Journal of computational physics, 84, 90-113, (1989) · Zbl 0677.65093 |

[9] | Cockburn, B.; Hou, S.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of computation, 54, 545-581, (1990) · Zbl 0695.65066 |

[10] | Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws V: multidimensional systems, Journal of computational physics, 141, 199-224, (1998) |

[11] | Cockburn, B.; Karniadakis, G.; Shu, C.-W., The development of discontinuous Galerkin methods, (), 3-50 · Zbl 0989.76045 |

[12] | Dumbser, M.; Munz, C.-D., Arbitrary high order discontinuous Galerkin schemes, (), 295-333 · Zbl 1210.65165 |

[13] | Dumbser, M.; Munz, C.-D., Building blocks for high order discontinuous Galerkin schemes, Journal of scientific computing, 27, 215-230, (2006) · Zbl 1115.65100 |

[14] | Taube, A.; Dumbser, M.; Balsara, D.; Munz, C.-D., Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations, Journal of scientific computing, 30, 441-464, (2007) · Zbl 1176.76075 |

[15] | Friedrichs, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, Journal of computational physics, 144, 194-212, (1998) · Zbl 1392.76048 |

[16] | Hesthaven, J.S.; Warburton, T., Nodal high-order methods on unstructured grids I. time-domain solution of maxwell’s equations, Journal of computational physics, 181, 186-221, (2002) · Zbl 1014.78016 |

[17] | Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of computational physics, 150, 97-127, (1999) · Zbl 0926.65090 |

[18] | Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065 |

[19] | Krivodonova, L.; Xin, J.; Remacle, J.-F.; Flaherty, J.E., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Applied numerical mathematics, 48, 323-338, (2004) · Zbl 1038.65096 |

[20] | Liu, X.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of computational physics, 115, 200-212, (1994) · Zbl 0811.65076 |

[21] | Qiu, J.; Shu, C.-W., Runge – kutta discontinuous Galerkin method using WENO limiters, SIAM journal on scientific computing, 26, 907-929, (2005) · Zbl 1077.65109 |

[22] | Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method: one dimensional case, Journal of computational physics, 193, 115-135, (2004) · Zbl 1039.65068 |

[23] | Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method: two dimensional case, Computers and fluids, 34, 642-663, (2005) · Zbl 1134.65358 |

[24] | W.H. Reed, T.R. Hill, Triangular mesh methods for neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. |

[25] | Shi, J.; Hu, C.; Shu, C.-W., A technique for treating negative weights in WENO schemes, Journal of computational physics, 175, 108-127, (2002) · Zbl 0992.65094 |

[26] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of computational physics, 77, 439-471, (1988) · Zbl 0653.65072 |

[27] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061 |

[28] | Stroud, A.H.; Secrest, D., Gaussian quadrature formulas, (1966), Prentice-Hall Inc. · Zbl 0156.17002 |

[29] | Suresh, A.; Huynh, H.T., Accurate monotonicity preserving scheme with runge – kutta time-stepping, Journal of computational physics, 136, 83-99, (1997) · Zbl 0886.65099 |

[30] | Takewaki, H.; Nishiguchi, A.; Yabe, T., Cubic interpolated pseudoparticle method (CIP) for solving hyperbolic equations, Journal of computational physics, 61, 261-268, (1985) · Zbl 0607.65055 |

[31] | Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, Journal of scientific computing, 17, 609-618, (2002) · Zbl 1024.76028 |

[32] | Torrilhon, M.; Balsara, D.S., High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems, Journal of computational physics, 201, 586-600, (2004) · Zbl 1076.76050 |

[33] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of computational physics, 54, 115-173, (1984) · Zbl 0573.76057 |

[34] | Yang, H.; schemes, An artificial compression method for ENO, The slope modification method, Journal of computational physics, 89, 125, (1990) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.