Very-high-order weno schemes

  • Authors:

  • G. A. Gerolymos;D. Sénéchal;I. Vallet

  • Affiliations:

  • Institut d'Alembert, Case 161, Université Pierre et Marie Curie (UPMC), 4 place Jussieu, 75005 Paris, France;Institut d'Alembert, Case 161, Université Pierre et Marie Curie (UPMC), 4 place Jussieu, 75005 Paris, France;Institut d'Alembert, Case 161, Université Pierre et Marie Curie (UPMC), 4 place Jussieu, 75005 Paris, France

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

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Abstract

We study weno(2r-1) reconstruction [D.S. Balsara, C.W. Shu, Monotonicity prserving weno schemes with increasingly high-order of accuracy, J. Comput. Phys. 160 (2000) 405-452], with the mapping (wenom) procedure of the nonlinear weights [A.K. Henrick, T.D. Aslam, J.M. Powers, Mapped weighted-essentially-non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys. 207 (2005) 542-567], which we extend up to weno17 (r=9). We find by numerical experiment that these procedures are essentially nonoscillatory without any stringent cfl limitation (cfl@?[0.8,1]), for scalar hyperbolic problems (both linear and scalar conservation laws), provided that the exponent p"@b in the definition of the Jiang-Shu [G.S. Jiang, C.W. Shu, Efficient implementation of weighted eno schemes, J. Comput. Phys. 126 (1996) 202-228] nonlinear weights be taken as p"@b=r, as originally proposed by Liu et al. [X.D. Liu, S. Osher, T. Chan, Weighted essentially nonoscillatory schemes, J. Comput. Phys. 115 (1994) 200-212], instead of p"@b=2 (this is valid both for weno and wenom reconstructions), although the optimal value of the exponent is probably p"@b(r)@?[2,r]. Then, we apply the family of very-high-order wenom"p"""@b"="r reconstructions to the Euler equations of gasdynamics, by combining local characteristic decomposition [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes iii, J. Comput. Phys. 71 (1987) 231-303], with recursive-order-reduction (ror) aiming at aleviating the problems induced by the nonlinear interactions of characteristic fields within the stencil. The proposed ror algorithm, which generalizes the algorithm of Titarev and Toro [V.A. Titarev, E.F. Toro, Finite-volume weno schemes for 3-D conservation laws, J. Comput. Phys. 201 (2004) 238-260], is free of adjustable parameters, and the corresponding rorwenom"p"""@b"="r schemes are essentially nonoscillatory, as @Dx-0, up to r=9, for all of the test-cases studied. Finally, the unsplit linewise 2-D extension of the schemes is evaluated for several test-cases.