Resolving Confusions over Third-Order Accuracy of Unstructured MUSCL

Abstract

In this paper, the authors discuss the unstructured MUSCL (U-MUSCL) reconstruction scheme, an unstructured-grid extension of the -scheme of Van Leer, proposed by Burg for the edge-based discretization. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers, but with confusions (e.g., third-order accuracy with or ). This paper clarifies some of these confusions: for example, the U-MUSCL scheme can be third-order accurate in the point-valued solution with on regular grids for linear equations in all dimensions, and it can be third-order accurate with as the QUICK scheme in one dimension. It is shown that the U-MUSCL scheme cannot be third-order accurate for nonlinear equations, except a very special case of on regular simplex-element grids, but it can be an accurate low-dissipation second-order scheme. It is also shown that U-MUSCL extrapolates a quadratic function exactly with on arbitrary grids, provided the gradient is computed by a quadratic least-squares method. Two techniques are discussed, which transform the U-MUSCL scheme into being genuinely third-order accurate on a regular grid: an efficient flux-reconstruction method and a special source-term quadrature formula for .