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In this paper, we discuss the U-MUSCL reconstruction scheme -- anunstructured-grid extension of Van Leer's kappa-scheme -- proposed by Burg forthe edge-based discretization [AIAA Paper 2005-4999]. This technique has beenwidely used in practical unstructured-grid fluid-dynamics solvers but withconfusions: e.g., third-order accuracy with kappa=1/2 or kappa=1/3. This paperclarifies some of these confusions: e.g., the U-MUSCL scheme can be third-orderaccurate in the point-valued solution with kappa=1/3 on regular grids forlinear equations in all dimensions, it can be third-order accurate withkappa=1/2 as the QUICK scheme in one dimension. It is shown that the U-MUSCLscheme cannot be third-order accurate for nonlinear equations, except a veryspecial case of kappa=1/2 on regular simplex-element grids, but it can be anaccurate low-dissipation second-order scheme. It is also shown that U-MUSCLextrapolates a quadratic function exactly with kappa=1/2 on arbitrary gridsprovided the gradient is computed by a quadratic least-squares method. Twotechniques are discussed, which transform the U-MUSCL scheme into beinggenuinely third-order accurate on a regular grid: an efficientflux-reconstruction method and a special source term quadrature formula forkappa=1/2.

Resolving Confusion Over Third Order Accuracy of U-MUSCL