Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes I. Volume integration

This work deals with the development of the direct reconstruction method (DRM) and its application to the volume integration of the discontinuous Galerkin (DG) method on multi-dimensional high-order mixed-curved meshes. The conventional quadrature-based DG methods require the humongous computational cost on high-order curved elements due to their non-linear shape functions. To overcome this issue, the flux function is directly reconstructed in the physical domain using nodal polynomials on a target space in a quadrature-free manner. Regarding the target space and distribution of the nodal points, DRM has two variations: the brute force points (BFP) and shape function points (SFP) methods. In both methods, one nodal point corresponds to one nodal basis function of the target space. The DRM-BFP method uses a set of points that empirically minimizes a condition number of the generalized Vandermonde matrix. In the DRM-SFP method, the conventional nodal points are used to span an enlarged target space of the flux function. It requires a larger number of reconstruction points than DRM-BFP but offers easy extendability to the higher-degree polynomial space and a better de-aliasing effect. A robust way to compute orthonormal polynomials is provided to achieve lower round-off errors. The proposed methods are validated by the 2-D/3-D Navier-Stokes equations on high-order mixed-curved meshes. The numerical results confirm that the DRM volume integration greatly reduces the computational cost and memory overhead of the conventional quadrature-based DG methods on high-order curved meshes while maintaining an optimal order-of-accuracy as well as resolving the flow physics accurately.N