Multidimensional upwinding: Its relation to finite elements

Abstract

Vertex‐based multidimensional upwind schemes for scalar advection are compared with shock‐capturing SUPG finite element methods based on linear triangular elements. Both methods share the same compact stencil and are formulated as cell‐wise residual distribution methods. The distribution for the finite element method is 1/3, supplemented with a Lax‐Wendrov‐type dissipation term, while the distribution for the upwind schemes is limited to the downstream nodes of the element. The multidimensional upwind schemes use positivity as the monotonicity criterion, while the finite element method includes a residual‐based non‐linear dissipation. For hyperbolic systems such as the compressible Euler equations the upwind method relies on a multidimensional wave model to decompose the residual into scalar contributions. From this observation a new SUPG formulation for systems is proposed in which the scalar SUPG method is applied to each of the decomposed residuals obtained from the wave model, thereby providing a better‐founded definition of the τ dissipation matrix and shock‐capturing term in the SUPG methods. Copyright © 1995 John Wiley & Sons, Ltd