A re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws

Abstract

A WENO re-averaging (or re-mapping) technique is developed that converts function averages on one grid to another grid to high order. Nonlinear weighting gives the essentially non-oscillatory property to the re-averaged function values. The new reconstruction grid is used to obtain a standard high order WENO reconstruction of the function averages at a select point. By choosing the reconstruction grid to include the point of interest, a high order function value can be reconstructed using only positive linear weights. The re-averaging technique is applied to define two variants of a classic CWENO3 scheme that combines two linear polynomials to obtain formal third order accuracy. Such a scheme cannot otherwise be defined, due to the nonexistence of linear weights for third order reconstruction at the center of a grid element. The new scheme uses a compact stencil of three solution averages, and only positive linear weights are used. The scheme extends easily to problems in higher space dimensions, essentially as a tensor product of the one-dimensional scheme. The scheme maintains formal third order accuracy in higher dimensions. Numerical results show that this CWENO3 scheme is third order accurate for smooth problems and gives good results for non-smooth problems, including those with shocks.

Introduction

Since their introduction in the late 1980s, essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) schemes [1], [2], [3], [4], [5], [6] have become two of the standard methods for approximating the solution to hyperbolic equations. They have been shown to be quite successful in a variety of applications. ENO and WENO schemes approximate the solution by a constant on each grid element, and advance in time explicitly by using a higher order polynomial reconstruction of the current solution. The ENO scheme computes several high order reconstructions, each on a different stencil, but uses only the stencil on which the solution is the smoothest (presumably the stencil does not cross a shock). The WENO schemes exploit the fact that often a convex linear combination of all the reconstructions can achieve even higher order accuracy than ENO at a specific point when the solution is smooth. A nonlinear weighting of the polynomial reconstructions, based on the solution smoothness, is actually used to avoid stencils that cross shocks in the solution.

Because reconstruction stencils need to avoid shocks, ENO and WENO are essentially upwind methods, and some form of flux-splitting is necessary when the system of equations is subject to flow in both directions. The central methods avoid flux splitting, and central WENO (CWENO) schemes were developed later in the 1990s [4], [6]. These methods use staggered grids, offset by one-half grid element, and so require higher order WENO reconstructions at the midpoint of a grid element.

Formally fifth and even ninth order CWENO (CWENO5 and CWENO9) schemes have appeared that use the classic WENO methodology to define high order reconstructions from low order ones [4], [6]. However, there is no classic CWENO3 scheme in the literature, due to the need for high order reconstruction at the center of the grid element. The linear weights required do not exist in the third (and seventh) order case. Indeed, quoting from Qiu and Shu [6, p. 194], “We remark that it is easy to verify that, when r is an even number ($r=2,4,\dots$), there are no linear weights that satisfy [the appropriate] condition…. Hence there are no third-, seventh-, eleventh-, etc., order central WENO reconstructions for the point values with these choices of stencils.” A classic CWENO3 scheme would combine two linear approximations, each requiring the solution on two grid elements, for an overall stencil of three grid elements. However, the two linear polynomials agree at the center point, and so no convex combination can be higher than second order in a classic WENO reconstruction.

Third order CWENO schemes have been proposed in the literature by Levy, Puppo, and Russo [7], [8]. We will call these schemes “augmented,” because they use a convex combination of the two linear approximations augmented directly by a higher order polynomial (a quadratic) in the case of one space dimension. Thus there is no weighting of low order polynomials to achieve higher order accuracy, as in classic WENO. Rather, high order accuracy comes directly from the quadratic reconstruction. One might view these methods as some kind of adaptive ENO schemes, because they choose the linear weights somewhat arbitrarily and use the WENO smoothness indicator to bias the stencil combination from the quadratic in smooth regions to one of the linears in non-smooth regions. To put it another way, classic WENO methodology uses lower order stencils, and weights them to increase accuracy in regions where the solution is smooth, as opposed to the augmented CWENO3 schemes, which include the higher-order stencil with the low order stencils and weights them to reduce accuracy near discontinuities. Clearly both approaches have a great deal of merit.

In this paper, we present a classic CWENO3 scheme that indeed combines only two linear polynomials. It also uses a compact stencil of three grid element values in the reconstruction. The key to this work is to develop a technique that allows us to reconstruct average solution values on a grid different from the original. In the case of CWENO3, two natural reconstruction grids are defined, one given by shifting the computational grid by one-half grid element and resizing by a factor of one-quarter locally, and the other is nonuniform and described later. Re-averaging (or re-mapping) the solution from the original computational grid to the new reconstruction grid is then accomplished using high order integration, as described in a recent paper [9] (see also [10], [11]). Once this re-averaging to the new grid is complete, we need a third order reconstruction of the solution at the endpoints of a reconstruction grid element, rather than at the center of a computational grid element. The linear weights always exist for the endpoints of a grid element, and so we can proceed to define a classic CWENO scheme that is formally third order accurate. A formally seventh order, and in fact any odd order, scheme could be devised with the same technique, although we do not present the details here (see, e.g., [12], [13] for very high order schemes).

When linear weights exist in a classic WENO reconstruction, they may still be problematic if some of them are negative [11]. Fortunately, Shi, Hu, and Shu provided a technique to handle this case [14]. Our new re-averaging technique provides an alternate way to deal with negative weights, by avoiding them altogether. One can re-average to a shifted and perhaps locally rescaled reconstruction grid that includes the targeted point as an endpoint, since the linear weights for the endpoint case are known to be positive.

Standard WENO reconstructions cannot be used for arbitrary points, since we may not have the required linear weights. However, in some cases one may wish to obtain formally high order approximate values at arbitrary points, say in a postprocessing step. Our re-averaging technique can be used to overcome this limitation. This is done in [15], wherein an Eulerian–Lagrangian (or semi-Lagrangian) scheme is devised that requires high-order reconstructions at arbitrary points.

We also present in this paper an extension of the scheme to problems in more than one space dimension. It turns out to be relatively straightforward to do so, in the sense that much of the computation remains the same as in the one dimensional case; that is, the scheme is developed in an essentially tensor product form, iterating the one space dimensional scheme. We therefore only present the case of two space dimensions. Since it is built on one-dimensional computations, the scheme is relatively straightforward to implement, and it is computationally efficient, as opposed to the multi-dimensional approach of Levy, Puppo, and Russo [7], [8]. We also prove that the scheme is indeed formally third order accurate.

The re-averaging WENO reconstruction technique is presented in the next section. Section 3 presents the new CWENO3 scheme for a scalar conservation equation in one space dimension. The scheme is extended to problems posed in two space dimensions in Section 4. Numerical results in one space dimension are presented in Section 5, and the technique is applied to the Euler system in Section 6. Numerical results in two space dimensions are presented in Section 7, and the paper ends with conclusions in the final section.