Adaptive-Quadrature Fluctuation-Splitting Schemes for the Euler Equations

In this paper, we consider fluctuation-splitting schemes that can capture an isolated shock over a suitably-oriented single triangular element and also recognize a rarefaction. A particular focus is on the evaluation of the fluctuation (or the cell-residual): a oneparameter-family quadrature rule is employed to evaluate the fluctuation, which endows the fluctuation with a wave recognition capability. The parameter value is chosen based on the nature of the nonlinear wave passing through the element, and then the resulting fluctuation is distributed to the nodes. This strategy, combined with various distribution schemes, defines a family of adaptive-quadrature fluctuation-splitting schemes. Results demonstrate the superior ability of the new schemes in handling nonlinear waves compared with standard fluctuation-splitting schemes which cannot capture shocks over a single element and also admits non-physical shocks unless some kind of entropy-fix is incorporated.