A study of multidimensional fifth-order WENO method for genuinely two-dimensional Riemann solver

In this paper we conduct a research on the multidimensional fifth-order reconstruction method for Balsara's genuinely two-dimensional (2D) HLLE Riemann solver. This solver possesses the multidimensional effect successfully by solving the 2D Riemann problem at the cell vertexes. However, it is difficult to be applied to higher-order reconstruction technology. In order to meet the requirement of the reconstruction of the variables at the vertexes, we construct the general WENO reconstruction formula to complete the reconstruction procedure along different spatial directions in turns. The Gaussian quadrature is employed to achieve the fifth-order accuracy of the numerical flux integration at the interface. Typical Simpson rule in Balsara's 2D HLLE solver is no longer used, and each Gaussian point is regarded as an independent 2D Riemann problem to obtain the 2D Riemann flux directly. Due to the inevitable numerical overshoots in the process of the fifth-order reconstruction, an accuracy-preserving limiter is introduced to enhance the computational stability, which could maintain the numerical accuracy in the smooth regions and suppress the overshoots near discontinuities. Solution accuracy tests show that the multidimensional WENO method could achieve the fifth-order accuracy. Several solution quality tests are also presented, which indicate that the method proposed in this study performs well in all test cases and can capture discontinuities and complex flow structures accurately and efficiently. • A multidimensional fifth-order reconstruction method is developed for Balsara's genuinely 2D HLLE Riemann solver. • This method uses the Gaussian quadrature approach to achieve the fifth-order accuracy for the numerical flux integration. • A new limiter adopting the discontinuity-detector technology is proposed to enhance the robustness.