Very-high-order WENO schemes

We study WENO(2r - 1) reconstruction [D.S. Balsara, CW. 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 is an element of (0.8, 1]), for scalar hyperbolic problems (both linear and scalar conservation laws), provided that the exponent p(beta) 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(beta) = 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(beta) = 2 (this is valid both for WENO and WENOM reconstructions), although the optimal value of the exponent is probably p(beta)(r) is an element of [2, r]. Then, we apply the family of very-high-order WENOMp beta=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 RORWENOMp beta=r schemes are essentially nonoscillatory, as Delta x -> 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. (C) 2009 Elsevier Inc. All rights reserved.