Very-high-order weno schemes

We study WENO(2r-1) reconstruction [Balsara D., Shu C.W.: J. Comp. Phys. 160 (2000) 405–452], with the mapping (WENOM) procedure of the nonlinear weights [Henrick A.K., Aslam T.D., Powers J.M.: J. Comp. 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 pβ in the definition of the Jiang-Shu [Jiang G.S., Shu C.W.: J. Comp. Phys. 126 (1996) 202–228] nonlinear weights be taken as pβ = r, as originally proposed by Liu et al. [Liu X.D., Osher S., Chan T.: J. Comp. Phys. 115 (1994) 200–212], instead of pβ = 2 (this is valid both for WENO and WENOM reconstructions), although the optimal value of the exponent is probably pβ(r) ∈ [2, r]. Then, we apply the family of very-high-order WENOMpβ=r reconstructions to the Euler equations of gasdynamics, by combining local characteristic decomposition [Harten A., Engquist B., Osher S., Chakravarthy S.R.: J. Comp. Phys. 71 (1987) 231–303], with recursiveorder-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 [Titarev V.A., Toro E.F.: J. Comp. Phys. 201 (2004) 238–260], is free of adjustable parameters, and the corresponding RORWENOMpβ=r schemes are essentially nonoscillatory, as ∆x → 0, up to r = 9, for all of the test-cases studied.