Very-high-order weno schemes

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)$(r=9)$. We find by numerical experiment that these procedures are essentially nonoscillatory without any stringent cfl limitation (cfl∈[0.8,1])$(\text{<small-caps>cfl</small-caps>}\in [0.8\text{,}1])$, for scalar hyperbolic problems (both linear and scalar conservation laws), provided that the exponent p_β$p_{\beta }$ 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_β=r$p_{\beta }=r$, 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_β=2$p_{\beta }=2$ (this is valid both for weno and wenom reconstructions), although the optimal value of the exponent is probably p_β(r)∈[2,r]$p_{\beta }(r)\in [2\text{,}r]$. Then, we apply the family of very-high-order wenom_pβ=r${\text{<small-caps>wenom</small-caps>}}_{p_{\beta }=r}$ 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 rorwenom_pβ=r${\text{<small-caps>rorwenom</small-caps>}}_{p_{\beta }=r}$ schemes are essentially nonoscillatory, as Δx→0$\mathrm{\Delta }x\to 0$, up to r=9$r=9$, for all of the test-cases studied. Finally, the unsplit linewise 2-D extension of the schemes is evaluated for several test-cases.