On the accuracy and efficiency of several discontinuous high-order formulations

Numerical accuracy and efficiency of several discontinuous high-order methods, including the quadrature based discontinuous Galerkin (QDG), nodal discontinuous Galerkin (NDG), spectral difference (SD) and correction procedure via reconstruction (CPR), for the conservation laws are analyzed and compared on both linear and curved quadrilateral elements. In the CPR formulation, both Lagrange-polynomial (LP) and chain-rule (CR) approaches are adopted to compute the flux divergence and their effects on the accuracy and efficiency of the resulting scheme are discussed. On linear quadrilateral elements, all the above schemes are one-dimensional in each natural coordinate direction. However, on curved elements, not all schemes can be reduced to a onedimensional form, although the SD and CPR formulations remain one-dimensional by design. The efficiency and accuracy of various formulations are compared on highly skewed curvilinear elements. Furthermore, a new approach to improve the accuracy on high order elements while not dramatically increasing the cost is developed. Several benchmark problems are simulated to further evaluate the performance of these schemes.