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

Journal of Computational Physics

Volumn 226, Issue 1, 2007, Pages 586-620

  Balsara, Dinshaw S ^a   Altmann, Christoph ^a,b   Munz, Claus Dieter ^b   Dumbser, Michael ^b  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Conservation laws; Higher order schemes; RKDG schemes; WENO schemes

Indexed keywords

GALERKIN METHODS;

CONSERVATION LAW; DISCONTINUOUS GALERKIN SCHEMES; ESSENTIALLY NON-OSCILLATORY SCHEMES; HERMITE; HIGH-ORDER SCHEME; RUNGE-KUTTA DISCONTINUOUS GALERKIN; RUNGE-KUTTUM DISCONTINUOUS GALERKIN SCHEME; SCALE-FREE; WEIGHTED ESSENTIALLY NON-OSCILLATORY SCHEME; WENO SCHEMES;

RUNGE KUTTA METHODS;

EID: 34548438386     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.04.032     Document Type: Article

References (34)

1. M. Abromowitz, I.E. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, vol. 55, 1964.
2. Balsara D.S., and Shu C.-W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics 160 (2000) 405-452
3. Balsara D.S., and Norton C. Highly parallel structured adaptive mesh refinement using parallel language-based approaches. Journal of Parallel Computation 27 (2001) 37-70
4. Balsara D.S. Second order accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophysical Journal Supplement 151 (2004) 149-184
5. Biswas R., Devine K.D., and Flaherty J. Parallel adaptive finite element methods for conservation laws. Applied Numerical Mathematics 14 (1994) 255-283
6. Burbeau A., Sagaut P., and Bruneau C.H. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational Physics 169 (2001) 111-150
7. Cockburn B., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Mathematics of Computation 52 (1989) 411-435
8. Cockburn B., Lin S.-Y., and 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 (1989) 90-113
9. Cockburn B., Hou S., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Mathematics of Computation 54 (1990) 545-581
10. Cockburn B., and Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. Journal of Computational Physics 141 (1998) 199-224
11. Cockburn B., Karniadakis G., and Shu C.-W. The development of discontinuous Galerkin methods. In: Cockburn B., Karniadakis G., and Shu C.-W. (Eds). Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I: Overview vol. 11 (2000), Springer 3-50
12. Dumbser M., and Munz C.-D. Arbitrary high order discontinuous Galerkin schemes. In: Cordier S., Goudon T., Gutnic M., and Sonnendrucker E. (Eds). Numerical Methods for Hyperbolic and Kinetic Problems. IRMA Series in Mathematics and Theoretical Physics (2005), EMS Publishing House, Zurich 295-333
13. Dumbser M., and Munz C.-D. Building blocks for high order discontinuous Galerkin schemes. Journal of Scientific Computing 27 (2006) 215-230
14. Taube A., Dumbser M., Balsara D., and Munz C.-D. Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations. Journal of Scientific Computing 30 (2007) 441-464
15. Friedrichs O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. Journal of Computational Physics 144 (1998) 194-212
16. Hesthaven J.S., and Warburton T. Nodal high-order methods on unstructured grids I. Time-domain solution of Maxwell's equations. Journal of Computational Physics 181 (2002) 186-221
17. Hu C., and Shu C.-W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics 150 (1999) 97-127
18. Jiang G., and Shu C.-W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics 126 (1996) 202-228
19. Krivodonova L., Xin J., Remacle J.-F., and Flaherty J.E. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Numerical Mathematics 48 (2004) 323-338
20. Liu X., Osher S., and Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics 115 (1994) 200-212
21. Qiu J., and Shu C.-W. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM Journal on Scientific Computing 26 (2005) 907-929
22. Qiu J., and 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 (2004) 115-135
23. Qiu J., and Shu C.-W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: two dimensional case. Computers and Fluids 34 (2005) 642-663
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., and Shu C.-W. A technique for treating negative weights in WENO schemes. Journal of Computational Physics 175 (2002) 108-127
26. Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77 (1988) 439-471
27. Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 83 (1989) 32-78
28. Stroud A.H., and Secrest D. Gaussian Quadrature Formulas (1966), Prentice-Hall Inc.
29. Suresh A., and Huynh H.T. Accurate monotonicity preserving scheme with Runge-Kutta time-stepping. Journal of Computational Physics 136 (1997) 83-99
30. Takewaki H., Nishiguchi A., and Yabe T. Cubic interpolated pseudoparticle method (CIP) for solving hyperbolic equations. Journal of Computational Physics 61 (1985) 261-268
31. Titarev V.A., and Toro E.F. ADER: Arbitrary high order Godunov approach. Journal of Scientific Computing 17 (2002) 609-618
32. Torrilhon M., and Balsara D.S. High order WENO schemes: investigations on non-uniform convergence for MHD Riemann problems. Journal of Computational Physics 201 (2004) 586-600
33. Woodward P., and Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics 54 (1984) 115-173
34. Yang H., and schemes A.a.c.m.f.E. the slope modification method. Journal of Computational Physics 89 (1990) 125