Very-high-order weno schemes

Journal of Computational Physics

Volumn 228, Issue 23, 2009, Pages 8481-8524

  Gerolymos, G A ^a   Senechal D ^a   Vallet, I ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Euler equations; High order schemes; Hyperbolic conservation laws; Smoothness indicators; weno schemes

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; PHYSICAL PROPERTIES; REPAIR;

ESSENTIALLY NON-OSCILLATORY; ESSENTIALLY NON-OSCILLATORY SCHEMES; EULER'S EQUATION; HIGH-ORDER; HIGH-ORDER SCHEME; HIGHER-ORDER; HYPERBOLIC CONSERVATION LAWS; ORDER REDUCTION; SMOOTHNESS INDICATORS; WENO SCHEMES;

EULER EQUATIONS;

EID: 70349748789     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.07.039     Document Type: Article

References (55)

1. Liu X.D., Osher S., and Chan T. Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115 (1994) 200-212
2. Jiang G.S., and Shu C.W. Efficient implementation of weighted eno schemes. J. Comput. Phys. 126 (1996) 202-228
3. Balsara D.S., and Shu C.W. Monotonicity prserving weno schemes with increasingly high-order of accuracy. J. Comput. Phys. 160 (2000) 405-452
4. Henrick A.K., Aslam T.D., and Powers J.M. Mapped weighted-essentially-non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207 (2005) 542-567
5. Titarev V.A., and Toro E.F. Finite-volume weno schemes for 3-D conservation laws. J. Comput. Phys. 201 (2004) 238-260
6. Toro E.F., and Titarev V. musta fluxes for systems of conservation laws. J. Comput. Phys. 216 (2006) 403-429
7. Borges R., Carmona M., Costa B., and Don W.S. An improved weno scheme for hyperbolic conservation laws. J. Comput. Phys. 227 (2008) 3191-3211
8. Pirozzoli S. Conservative hybrid compact-weno schemes for shock-turbulence interaction. J. Comput. Phys. 178 (2002) 81-117
9. Ponziani D., Pirozzoli S., and Grasso F. Development of optimized weno schemes for multiscale compressible flows. Int. J. Numer. Method Fluid 42 (2003) 953-977
10. Martín M.P., Taylor E.M., Wu M., and Weirs V.G. A bandwidth-optimized weno scheme for effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220 (2006) 270-289
11. Taylor E.M., Wu M., and Martín M.P. Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. J. Comput. Phys. 223 (2007) 384-397
12. Harten A., Engquist B., Osher S., and Chakravarthy S.R. Uniformly high-order accurate essentially nonoscillatory schemes iii. J. Comput. Phys. 71 (1987) 231-303
13. Johnsen E., and Colonius T. Implementation of weno schemes in compressible multicomponent flow problems. J. Comput. Phys. 219 (2006) 715-732
14. Suresh A., and Huynh H.T. Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys. 136 (1997) 83-99
15. Qiu J., and Shu C. Hermite weno schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: 1-D case. J. Comput. Phys. 193 (2003) 115-135
16. Qiu J., and Shu C. Hermite weno schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: 2-D case. Comput. Fluid 34 (2005) 642-663
17. Capdeville G. A hermite upwind weno scheme for solving hyperbolic conservation laws. J. Comput. Phys. 227 (2008) 2430-2454
18. Capdeville G. Towards a compact high-order method for nonlinear hyperbolic systems - II: The Hermite hllc scheme. J. Comput. Phys. 227 (2008) 9428-9462
19. Toro E.F., and Titarev V. ader schemes for scalar nonlinear hyperbolic conservation laws with source-terms in 3D. J. Comput. Phys. 202 (2005) 196-215
20. Titarev V.A., and Toro E.F. ader schemes for 3D nonlinear hyperbolic systems. J. Comput. Phys. 204 (2005) 715-736
21. Toro E.F., and Titarev V. Derivative Riemann solvers for systems of conservation laws and ader methods. J. Comput. Phys. 212 (2006) 150-165
22. Harten A., and Osher S. Uniformly high-order accurate nonoscillatory schemes i. SIAM J. Numer. Anal. 24 (1987) 279-309
23. Shu C.W., and Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439-471
24. Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7 (1954) 159-193
25. Sod G.A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27 (1978) 1-31
26. Woodward P., and Colella P. The numerical simulation of 2-D fluid flow with strong shocks. J. Comput. Phys. 54 (1984) 115-173
27. Shu C.W., and Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes II. J. Comput. Phys. 83 (1989) 32-78
28. Shu C.W. eno and weno schemes for hyperbolic conservation laws. In: Cockburn B., Johnson C., Shu C.W., and Tadmor E. (Eds). Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics vol. 1697 (1998), Springer, Berlin [ 325-432 (also NASA CR-97-206253 and ICASE-97-65 Rep., NASA Langley Research Center, Hampton [VA, USA])
29. Lax P.D., and Liu X.D. Solution of 2-D Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19 (1998) 319-340
30. Zorich V.A. Mathematical Analysis II (2004), Springer, Heidelberg [
31. G.A. Gerolymos, D. Sénéchal, I. Vallet, Performance of very-high-order upwind schemes for dns of compressible wall-turbulence, Int. J. Numer. Method Fluid, in press, doi:10.1002/fld.2096. Available online 2 July 2009.
32. Van Leer B. Upwind and high-resolution methods for compressible flow: from donor cell to residual distribution schemes. Commun. Comput. Phys. 1 2 (2006) 192-206 (also AIAA Paper 2003-3559)
33. Van Leer B. Towards the ultimate conservative difference scheme - v - a 2-order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136
34. maxima, A Computer Algebra System. .
35. Gerolymos G.A., and Bréus J.P. Computation of unsteady nozzle flow resulting from fluctuating back-pressure using Euler equations. Aerosp. Sci. Technol. 2 2 (1998) 91-105
36. Butcher J.C. General linear methods. Acta Numer. 15 (2006) 157-256
37. Gottlieb S., Shu C.W., and Tadmor E. Strong stability preserving high-order time discretization methods. SIAM Rev. 43 1 (2001) 89-112
38. Gottlieb S., and Gottlieb L.A.J. Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coefficient operators. J. Sci. Comput. 18 1 (2003) 83-109
39. Gottlieb S. On high order strong stability preserving Runge-Kutta and multistep time discretizations. J. Sci. Comput. 25 1 (2005) 105-128
40. Ruuth S.J. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Math. Comput. 75 223 (2005) 183-207
41. M.L. Overton, Numerical Computing with ieee Floating Point Arithmetic, SIAM, Philadelphia, PA, USA.
42. R. Courant, K.O. Friedrichs, Supersonic Flow and Shock-Waves, Springer, New York, NY, USA, 1948 (corrected 5, printing 1999 edition).
43. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Springer-Verlag, Berlin [
44. Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357-372
45. Hu C., and Shu C.W. Weighted essentially nonoscillatory schemes on triangular meshes. J. Comput. Phys. 150 (1999) 97-127
46. Zhang Y.T., and Shu C.W. WENO3 scheme for 3-D tetrahedral meshes. Commun. Comput. Phys. 5 (2009) 836-848
47. Dumbser M., Balsara D.S., Toro E.F., and Munz C.D. A unified fremework for the construction of 1-step finite-volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227 (2008) 8209-8253
48. A. Rohde, Eigenvalues and Eigenvectors of the Euler Equations in General Geometries, AIAA Paper 2001-2609, 2001.
49. Shu C.W. tvb uniformly high-order schemes for conservation laws. Math. Comput. 29 (1987) 105-121
50. Zhang T., and Zheng Y. Conjecture on the structure of the solutions of the Riemann problem for 2-D gas dynamics systems. SIAM J. Math. Anal. 21 3 (1990) 593-630
51. Schulz-Rinne C.W. Classification of the Riemann problem for 2-D gas dynamics. SIAM J. Math. Anal. 24 1 (1993) 76-88
52. Schulz-Rinne C.W., Collins J.P., and Glaz H.M. Numerical solution of the Riemann problem for 2-D gas dynamics. SIAM J. Sci. Comput. 14 6 (1993) 1394-1414
53. Kurganov A., and Tadmor E. Solution of 2-D Riemann problems of gas dynamics without Riemann solvers. Numer. Method Part. Diff. Eq. 18 (2002) 584-608
54. Cockburn B., and Shu C.W. The Runge-Kutta discontinuous Galerkin method for conservation laws - V. J. Comput. Phys. 141 (1998) 199-224
55. Shi J., Hu C., and Shu C.W. A technique for treating negative weights in weno schemes. J. Comput. Phys. 175 (2002) 108-127