We explore how the weighted average flux approach can be used to generate first-and second-order accurate finite volume schemes order accuracy can be achieved by solving the conventional for the linear advection equatons in one, two, and three space dipiecewise constant Riemann problem as in the first-order mensions. The derived schemes have multidimensional upwinding Godunov method; no reconstruction/evolution steps are aspects and good stability properties. From the two-dimensional necessary, although these processes may also be admitted.
methods, we construct a scheme for nonlinear systems of hyper-
The accuracy comes from utilizing this solution averaged bolic conservation laws that is second-order accurate in smooth flow. Spurious oscillations are controlled by making use of oneover space and time. This averaging takes the form of an dimensional TVD limiter functions. Numerical results are presented integral of the flux, or chosen variables, over some volume. for the shallow water equations in two space dimensions. The equiv-The WAF method has been extended for use with large alent schemes are derived for nonlinear systems in three space timesteps (up to a Courant number of 2) [37,36] and for dimensions. แฎ 1997 Academic Press use on moving grids [5].
The standard way to extend these schemes to two and
1. Introduction
three space dimensions is via space operator splitting, as discussed by Strang [28], in which the one-dimensional Upwind methods for computational fluid dynamics difference operators are used in each dimension in turn. (CFD) form a respectable class of numerical techniques This approach has been shown to be successful: see, for available to the CFD practitioner today. This is the result example, the review paper by Woodard and Colella [45]. of an intensive research activity spanned over many years. An important issue is how to extend Godunov-type The distinguished works of Godunov [9], van Leer [40, schemes to two and three dimensions without space split-41], Roe [20, 22], Osher and Solomon [18], Harten [11], ting. One approach is to use solutions to one-dimensional and many others, have provided a solid theoretical framegrid-aligned Riemann problems and one-dimensional opwork for further advancement.
erators to construct multidimensional finite volume An important issue is how to generalize the first-order schemes. This approach has been used to construct the Godunov method [9] to second or higher order accuracy. MUSCL-Hancock scheme [19]. A similar approach has Van Leer [40, 41] proposed his monotone upwind schemes been used by LeVeque [16,17] and Colella [8] to generate for conservation laws (MUSCL) approach whereby the finite volume schemes for nonlinear systems, although they piecewise constant cell average states in the Godunov account for at least some aspects of multidimensional wave method are replaced by reconstructed states that admit propagation in the construction of their schemes, and as spacial variation within each cell. A class of second-order a result their schemes have better stability properties than Godunov-type methods based on this approach have been the MUSCL-Hancock scheme, as will be discussed later. constructed. Examples are the PLM method of Colella [7], A higher level of upwinding for multidimensional systems the GRP method of Ben-Artzi and Falcovitz [2] and the of equations is achieved by the so-called multidimensional MUSCL-Hancock scheme [42]. An alternative approach upwind methods. See, for example, Baines [1] or Roe [25, for constructing second-order Godunov type methods is 24]. A good review of multidimensional upwinding has the weighted average flux (WAF) approach [31]. Its origins been written by van Leer [43], in which many more refergo back to a random flux approach [30] which was later ences can be found. proved to be second-order accurate in a statistical sense
In this paper we discuss how the WAF approach can be used to generate finite volume schemes in more than one by Toro and Roe [39]. The WAF approach has been shown 1