Toward Robust and Efficient High-Order Summation-By-Parts Discretizations for Compressible Viscous Flows on Unstructured Simplicial Meshes

This thesis presents several advancements that contribute to the development of robust, efficient, and flexible high-order methods for computational fluid dynamics (CFD). The underpinning tool in these advancements is the summation-by-parts (SBP) framework along with simultaneous approximation terms (SATs). The SBP-SAT method enables the construction of provably stable discretizations of the governing equations of fluid dynamics, thereby enhancing the robustness of CFD algorithms. Additionally, the application of this approach on unstructured simplicial meshes facilitates automatic mesh generation, providing flexibility to handle complex geometries. Relative to some of the popular high-order methods, the entropy-stable SBP-SAT scheme on simplicial meshes is computationally expensive. To mitigate this drawback, we have made advancements in three key areas: the development of efficient entropy-stable viscous SATs, the development of alternative entropy-stable formulation, and the construction of novel SBP operators. We establish that the existing viscous SAT framework can be used to construct entropy-stable discretizations of the Navier--Stokes equations, enabling the use of entropy-stable SATs that couple only immediate neighboring elements. Furthermore, the viscous SAT framework is extended to encompass a wider class of SATs, and functional superconvergence is shown for the SATs that satisfy primal and adjoint consistency conditions. Building upon the entropy-split formulation primarily used within the traditional finite-difference framework, we develop an entropy-split entropy-stable discretization for element-type SBP-SAT schemes. The method offers substantial improvements in terms of efficiency compared to the Hadamard-form entropy-stable discretizations based on two-point flux functions. Furthermore, a locally conservative hybrid formulation of the scheme is proposed to mitigate the loss of conservation. Finally, a new matrix-type interface dissipation operator is developed to enable the use of the entropy-split entropy-stable scheme for problems with discontinuities. To further enhance the efficiency of the entropy-stable SBP-SAT method, several novel SBP operators are constructed. Construction of efficient SBP operators necessitate the derivation of high-order quadrature rules with minimal node count. We derived symmetric quadrature rules with positive weights that include boundary nodes as well as those with exclusively interior nodes, extending the available sets of quadrature rules to higher degrees than currently available while also maintaining high level of quadrature efficiency. Finally, we introduced a novel approach to construct SBP operators on simplices with a tensor-product structure. The accuracy and sparsity of these tensor-product split-simplex operators substantially enhance the efficiency of SBP-SAT discretizations on simplicial meshes.

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