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Multidimensional upwinding: Its relation to finite elements. (English) Zbl 0840.76032

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-Wendroff-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 nonlinear dissipation. 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 \(\tau\) dissipation matrix and shock-capturing term in the SUPG methods.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76L05 Shock waves and blast waves in fluid mechanics
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