Very-high-order weno schemes

We study weno(2 r − 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 β 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 β = 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 β = 2 (this is valid both for weno and wenom reconstructions), although the optimal value of the exponent is probably p β ( r ) ∈ [ 2 , r ] . Then, we apply the family of very-high-order wenom p β = 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 β = r schemes are essentially nonoscillatory, as Δ 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.