| While second order methods are dominant in most compressible flow simulations,many types of problems, such as computational aeroacoustics (CAA), vortex-dominant flows and large eddy simulation (LES) of turbulent flows, call for higherorder accuracy. The main deficiency of widely available, second-order methods forthe accurate simulations of the above-mentioned flows is the excessive numericaldiffusion and dissipation of vorticity. Applications of high-order accurate, low-diffusion and low dissipation numerical methods can significantly alleviate thisdeficiency of the traditional second order methods, and improve predictions ofvortical and other complex, separated, unsteady flows. Therefore, various high-ordermethods have been developed in the last two decades.As the leader of high-order numerical methods for compressible flowcomputations in aerospace applications, the discontinuous Galerkin (DG) method hasrecently become popular for problems with both complex physics and geometry. TheDG method was originally developed by Reed and Hill to solver the neutron transportequation. The development of high-order DG methods for hyperbolic conservationlaws was pioneered by Cockburn, Shu and other collaborators in a series of papers onthe Runge-Kutta DG (RKDG) method. It is well known that the DG method has manydistinguished features: (1) It is well suited for complex geometries since it can beapplied on unstructured and hybrid grids; (2) It is compact, as each element isdependent on its immediate face-neighbors. This feature makes the DG methodsideally suitable for parallel computing. (3) It can easily handle adaptive strategies,since refining or coarsening an element can be achieved without considering thecontinuity restriction commonly associated with conforming elements. (4) It hasseveral useful mathematical properties with respect to conservation, stability andconvergence.However, the DG method does have a number of weaknesses, including the hugememory required and high computational cost. In an implicit DG method, the blockdiagonal matrix requires a storage of (ndof*neqs)*(ndof*neqs)*nelems, where ndof isthe number of degrees of freedom (DOFs) per equation, neqs is the number ofcomponents in the solution vector and nelems is the total number of cells in the grid.For example, the storage of this block diagonal matrix alone requires 10,000 words perelement for a fourth-order DG scheme in 3D! Indeed, the lack of efficient solver isone of the bottlenecks in the development of the DG method for solving realisticproblems.Comparing with the traditional second order DG method, the second order finitevolume (FV) and finite difference (FD) methods need less amount of memory and computational cost on the same mesh, because they do not need to compute thevolume integrals and the additional equations for the DOFs corresponding to thederivatives. However, for high-order FV reconstructions, large grid stencil should beadopted, which means that tremendous memory and CPU time are required either.As discussed above, all the above-mentioned high-order methods have theirdistinguished features, and there is still room for improvement. Therefore, a naturalchoice is to combine their advantageous features in terms of efficiency in bothmemory and CPU time for 3D realistic complex configurations. In this paper, bycomparing the DG and the FV methods, the concept of'static reconstruction'and'dynamic reconstruction'are proposed for high-order numerical methods. Based onthe new concept, a class of hybrid DG/FV methods is presented for one- and two-dimensional conservation law using a'hybrid reconstruction'approach. In the hybridDG/FV schemes, the lower-order derivatives of the piecewise polynomial arecomputed locally in a cell by the traditional DG method (called as'dynamicreconstruction'), while the higher-order derivatives are re-constructed by the'staticreconstruction'of the FV method, using the known lower-order derivatives in the cellitself and in the immediate neighbour cells. The hybrid DG/FV method can reduce theCPU time and memory requirement greatly than the traditional DG method with thesame order of accuracy, and can be extended directly to unstructured and hybrid gridsin two and three dimensions similar to the DG and/or FV methods. The hybrid DG/FVmethod is applied to one- and two-dimensional conservation law, including linear andnonlinear scalar equation and Euler equations. In order to capture the strong shockwaves without spurious oscillations, a simple shock detection approach is developedto mark'trouble cells', and a new vertex Hermit WENO limiter is proposed for high-order schemes. The numerical results demonstrate the accuracy. In addition, byanalyzing the eigenvalues of the semi-discretized system in one dimension, we discussthe spectral properties of the hybrid DG/FV schemes.This dissertaion is divided into seven chapters as follows:The first chapter is the introduction, in which the progress in high-order methodsare reviewed briefly, including the finite difference method on structured grid, such asWENO schemes, compact scheme, WCNS, and the methods on unstructured gridbased on integral algorithm, such as the finite volume method, the discontinuousGalekin method, etc. In addition, the grid generation techniques, especially the hybridgrid techniques are summarized. Finally, the main work of this dissertaion isintroduced briefly.In the second chapter, the fundamental of DGM is presented in detail. In theframework of RKDG finite element method proposed by Cockburn and Shu, we adopta DG method based on Taylor basis functions developed by Luo Hong. Unlike the traditional DG methods, in which either standard Lagrange finite element orhierarchical node-based basis functions are used to represent numerical polynomialsolutions in each cell, this DG method represents the numerical polynomial solutionsusing a Taylor series expansion at the centroid of cell. Also, Taylor basis functions arehierarchical, which is convenient to handle p- or hp-multigrid strategy.In the third chapter, the hybrid DG/FV schemes are discussed. By comparingDGM and FVM, the concepts of'static reconstruction'and'dynamic reconstruction'are proposed for high-order numerical schemes. Based on the new concept of'hybridreconstruction', a new class of hybrid DG/FV schemes is presented for unstructuredgrids to solve the 2D conservation law.In Chapter Four, some typical cases are tested for smooth solutions in one- andtwo-dimension. The numerical results demonstrate the desired order of accuracy andthe super-convergence property is shown for the third-order hybrid DG/FV schemes inone-dimension. Also, by analyzing eigenvalue of the semi-discretized system in one-dimension scalar conservation law, we have studied the spectral behavior of presentDG/FV schemes, and the spectral super-convergence property is found.We discuss the shock detection and the limiter for discontinuous problems inChapter Five. Based on analyzing the difference of variables between the left and theright sides of grid interface in smooth region and near shock waves, a shock detectionapproach is developed to mark the'trouble cells'in discontinuity region, and appliedin high-order discontinuous Galerkin methods to suppress spurious oscillations neardiscontinuities coupled with proper limiters.In the sixth chapter, the hybrid DG/FV schemes are validated by some typicalone-dimensional cases, including the Lax, Sod, Shu's problems, and two-dimensionaltransnic flows over the NACA0012 airfoil and the double Mach reflection case withstrong inclined shock wave movement. The numerical results demonstrate that thehybrid DG/FV schemes have good capability of capturing shock waves and contact-discontinuity with high resolution.The conclusions are presented in the last chapter. The work of this dissertation issummarized, and the possible future work on the hybrid DG/FV schemes is discussedalso.
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