Efficient dimension-by-dimension higher order finite-volume methods for a Cartesian grid with cell-based refinement

A new higher-order scheme for hyperbolic systems of conservation law on Cartesian grids with cell-based refinement is proposed. The scheme is based on the finite volume methods, which admits a simple formulation around the hanging-nodes. Dimension-by-dimension reconstruction is applied to utilize the advantage of Cartesian grids. A quadrature modification flux (QMF) is introduced, which modifies the second-order error term of the flux integration over the cell-interface. The QMF can be evaluated using the state variables and the first derivatives stored in the face-left and right cells. There is no need to calculate the point-value or to use multi-points flux quadrature. In addition, a two-step reconstruction method is introduced to achieve a higher-order reconstructed value. The scheme achieves fourth-order accuracy where the grids are locally uniform, and retains second-order accuracy around the hanging-nodes. The accuracy and efficiency of the scheme is demonstrated in some example problems: inviscid vortex advection, Shu-Osher problem, double Mach reflection, and transonic flow around the NACA 0012 airfoil. The present scheme is demonstrated to function stably and consistently.