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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. • As a quadrature-free method, the DRM framework is proposed and applied to the volume integration of DG methods for the Navier-Stokes equations. • According to the target space and interpolation nodes, DRM-BFP and DRM-SFP are developed on multi-dimensional higher-order mixed-curved mesh. • Through numerical analyses and computations, efficiency and accuracy of DRM are extensively validated on higher-order mixed curved mesh. [ABSTRACT FROM AUTHOR]

Abstract The present paper deals with a new improvement of hierarchical multi-dimensional limiting process for resolving the subcell distribution of high-order methods on two-dimensional mixed meshes. From previous studies, the multi-dimensional limiting process (MLP) was hierarchically extended to the discontinuous Galerkin (DG) method and the flux reconstruction/correction procedure via reconstruction (FR/CPR) method on simplex meshes. It was reported that the hierarchical MLP (h MLP) shows several remarkable characteristics such as the preservation of the formal order-of-accuracy in smooth region and a sharp capturing of discontinuities in an efficient and accurate manner. At the same time, it was also surfaced that such characteristics are valid only on simplex meshes, and numerical Gibbs–Wilbraham oscillations are concealed in subcell distribution in the form of high-order polynomial modes. Subcell Gibbs–Wilbraham oscillations become potentially unstable near discontinuities and adversely affect numerical solutions in the sense of cell-averaged solutions as well as subcell distributions. In order to overcome the two issues, the behavior of the h MLP on mixed meshes is mathematically examined, and the simplex-decomposed P1 -projected MLP condition and smooth extrema detector are derived. Secondly, a troubled-boundary detector is designed by analyzing the behavior of computed solutions across boundary-edges. Finally, h MLP_BD is proposed by combining the simplex-decomposed P1 -projected MLP condition and smooth extrema detector with the troubled-boundary detector. Through extensive numerical tests, it is confirmed that the h MLP_BD scheme successfully eliminates subcell oscillations and provides reliable subcell distributions on two-dimensional triangular grids as well as mixed grids, while preserving the expected order-of-accuracy in smooth region. Highlights • Subcell monotonicity and its extension on 2-D mixed meshes have been realized. • Novel troubled-boundary detector is designed to capture subcell G–W oscillations. • h MLP_BD limiting is validated by extensive test cases on 2-D mixed meshes. [ABSTRACT FROM AUTHOR]

• DRM has been extended to DG surface integration of Euler/Navier-Stokes equations. • Adaptive MGS process is designed for optimal physical bases of restricted space. • DRM has been verified to significantly reduce computing cost and memory overhead. As the second extension of the direct reconstruction method (DRM), DRM for the surface integration of the discontinuous Galerkin (DG) weak formulation is presented on multi-dimensional high-order mixed meshes. From the previous study, DRM was successfully extended to the volume integration of the DG method, and produced a significant reduction in both the computational cost and memory overhead, particularly for high-order solution approximations on 3-D high-order meshes. DRM has the potential to reduce the computational cost and memory overhead further by treating the surface integration term consistently. To realize this, we design a linear dependency detector and formulate an adaptive orthonormalization process from the modified Gram-Schmidt (MGS) process. DRM is then applied to the restricted function space of the surface integration in an adaptive and optimal manner. The resulting methods, namely the DRM surface integration with the brute force points (BFP) and shape function points (SFP) methods, perform very well on high-order mixed-curved meshes. In addition, the adaptive MGS process is robust and accurate in extracting linear faces/edges from a given set of mixed-curved meshes. Various benchmark tests from the compressible Euler and Navier-Stokes equations verify the impact of DRM for 3-D large-scale computations on mixed-curved meshes. As an example, the DRM framework applied to the 3-D surface and volume integrations with DG- P 3 approximation on P 3-mesh may yield O (10) × speed-up, compared to conventional quadrature-based methods. [ABSTRACT FROM AUTHOR]