Resolving Confusion Over Third Order Accuracy of U-MUSCL

TL;DR

This paper clarifies the third-order accuracy conditions of the U-MUSCL scheme, showing its capabilities and limitations for linear and nonlinear equations on various grids, and proposes methods to achieve genuine third-order accuracy.

Contribution

It provides a detailed analysis of the U-MUSCL scheme's accuracy, resolving confusions, and introduces techniques to attain true third-order accuracy on regular grids.

Findings

U-MUSCL can be third-order accurate with kappa=1/3 on regular grids for linear equations.

U-MUSCL can be third-order accurate with kappa=1/2 as the QUICK scheme in 1D.

The scheme is generally second-order for nonlinear equations, except in special cases.

Abstract

In this paper, we discuss the U-MUSCL reconstruction scheme -- an unstructured-grid extension of Van Leer's kappa-scheme -- proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999]. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers but with confusions: e.g., third-order accuracy with kappa=1/2 or kappa=1/3. This paper clarifies some of these confusions: e.g., the U-MUSCL scheme can be third-order accurate in the point-valued solution with kappa=1/3 on regular grids for linear equations in all dimensions, it can be third-order accurate with kappa=1/2 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 kappa=1/2 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 kappa=1/2 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 kappa=1/2.