Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Abstract

Central WENO reconstruction procedures have shown very good performances in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and more space dimensions, on different types of meshes. Their most recent formulations include WENOZ-type nonlinear weights, but in this context a thorough analysis of the definition of the global smoothness indicator τ is still lacking. In this work we first prove results on the asymptotic expansion of multi-dimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of CWENOZ schemes also beyond those considered in this paper. Next, we introduce the optimal definition of τ for the one-dimensional CWENOZ schemes and for one example of two-dimensional CWENOZ reconstruction. Numerical experiments of one and two dimensional test problems show the good performance of the new schemes.Keywords. Central WENOZ (CWENOZ) essentially non-oscillatory reconstructions finite volume schemes smoothness indicators MSC2010. 65M08 65M20In this section we prove the theoretical results to analyse the CWENO and CWENOZ reconstruction procedures in one and more space dimensions. To this end, we restrict here to the scalar case, since usually, in the case of systems of conservation laws, the reconstruction procedures are applied component-wise, directly to the conserved variables or after the local characteristic projection.

Efficient Implementation of Adaptive Order Reconstructions

“…, r 2 ; r 1 ) by Balsara, Garain and Shu [4]. For l = 2 they essentially coincide with CWENOZ reconstructions of [14], and for l > 2 the reconstruction is a blend of the reconstruction polynomials given by WENO-AO(r k ; r 1 ) for k = 2, . .…”

“…Applications to stochastic Galerkin have been considered in [18] and to Hamilton-Jacobi equations in [36].One of the advantages of this approach is the possibility to achieve genuinely multi-dimensional reconstructions that do not rely on dimensional splitting and that are, theoretically and in practice, not more challenging than the one-dimensional counterpart (see e.g. [29] for AMR grids and [38,17,16] for simplicial ones).CWENO reconstructions of arbitrary high orders have been studied in [12] and the use of Zweights has been pursued in [11,14].A very important result of [14] is an analysis of the multidimensional oscillation indicators, leading to general results that support the design of Z-weights at arbitrary order for very general one-and multi-dimensional finite volume grids.The optimal convergence rate on smooth data can be easily achieved when the degree gap between the central high order and the lower degree polynomials is not too high (see [12,14]). Thus, in the design of very high order reconstructions, one has to employ "low degree" polynomials whose stencils are still quite large and that, as a consequence, are not very good at avoiding discontinuities in complex multi-dimensional flows.…”

“…CWENO reconstructions of arbitrary high orders have been studied in [12] and the use of Zweights has been pursued in [11,14].A very important result of [14] is an analysis of the multidimensional oscillation indicators, leading to general results that support the design of Z-weights at arbitrary order for very general one-and multi-dimensional finite volume grids.…”

“…The optimal convergence rate on smooth data can be easily achieved when the degree gap between the central high order and the lower degree polynomials is not too high (see [12,14]). Thus, in the design of very high order reconstructions, one has to employ "low degree" polynomials whose stencils are still quite large and that, as a consequence, are not very good at avoiding discontinuities in complex multi-dimensional flows.…”

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

“…In this way, they are not able to detect truly multidimensional singularities that are not alligned with the axis. Our goal is to develop genuinely multidimensional smoothness indicators and prove their properties in the 2D setting (for genuinely multidimensional smoothness indicators, see also [12]). Moreover, we will show how these indicators are useful in the approximation of a class of first order time-dependent Hamilton-Jacobi (HJ) equations, in the form v t + H(x, y, v x , v y ) = 0, (t, x, y) ∈ [0, T ] × R 2 , v(0, x, y) = v 0 (x, y),…”