A new method of obtaining third-order accuracy on unstructured grid flow solvers is presented. The method involves a simple correction to a traditional linear Galerkin scheme on tetrahedra and can be conveniently added to existing second-order accurate node-centered flow solvers. The correction involves gradients of the flux computed with a quadratic least squares approximation. However, once the gradients are computed, no second derivative information or high-order quadrature is necessary to achieve third-order accuracy. The scheme is analyzed both analytically using truncation error, and numerically using solution error for an exact solution to the Euler equations. Two demonstration cases for steady, inviscid flow reveal increased accuracy and excellent shock capturing with no loss in steady-state convergence rate. Computational timing results are presented which show the additional expense from the correction is modest compared to the increase in accuracy.

No abstract

A novel high-order method, termed flux correction, previously formulated for inviscid flows, is extended to viscous flows on arbitrary triangular grids. The correction method involves the addition of truncation error-canceling terms to the second-order linear Galerkin (node-centered finite volume) scheme to produce a third-order inviscid and fourth-order viscous scheme. The correction requires minimal modification of the underlying secondorder scheme. As such, the method retains many of the advantages of traditional finite volume schemes, including robust shock capturing, low algorithmic complexity, and solver efficiency. In addition, we extend the scheme to unsteady flows. Verification and validation studies in two dimensions are presented. Significant improvement in accuracy is observed in all cases with tolerable computational and algorithmic overhead.

No abstract

No abstract