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Development and Investigation of Accurate High-Order Generalized Summation-by-Parts Discretizations for Computational Fluid Dynamics

AMS Seminar Series

Speaker: David Craig Penner, University of Toronto

Presentation

Seminar Slide Deck (PDF-2.6MB)

Abstract

The numerical solution of the equations governing turbulent fluid flows, whether the Reynolds-averaged Navier-Stokes equations or solely the Navier-Stokes equations, as in direct and large-eddy simulations, is computationally expensive. High-order methods have the potential to reduce this computational cost by providing higher accuracy per degree of freedom relative to low-order schemes. Spatial discretization schemes based on the generalized summation-by-parts property have been developed in recent years as a general approach for designing high-order numerical methods. This work delineates how to obtain accurate solutions and, particularly, superconvergent integral functionals when solving linear and nonlinear partial differential equations governing computational fluid dynamics problems of increasing practical relevance. The specific focus is on numerical schemes constructed on block-structured grids, where the spatial derivatives in the governing equations are approximated with high-order tensor-product generalized summation-by-parts operators.

To begin, various components of high-order flow solvers based on generalized summation-by-parts operators are developed including novel artificial dissipation operators and two approaches for high-order grid generation and refinement for traditional and element-type operators. Next, it is shown that functional superconvergence is retained for generalized summation-by-parts discretizations of the linear convection equation in curvilinear coordinates. Furthermore, four dual-consistent discretizations of the two-dimensional linear convection equation based on the mortar-element and global summation-by-parts-operator approaches are derived and characterized in terms of truncation error, solution accuracy, and functional accuracy.

Finally, using information gained from the analysis of the linear convection equation, a generalized summation-by-parts discretization for obtaining superconvergent functionals when solving sufficiently smooth problems governed by the Euler equations is proposed. Furthermore, the key features of a given discretization having an impact on solution and functional accuracy are delineated and analyzed. These features are identified to include the representation of the geometry, the approximation of the metrics, and the approximation of the wall normal in the flow tangency boundary condition.

Biography

Photo of David Craig Penner
David Craig Penner is a Ph.D. Candidate in Aerospace Engineering who is interested in the development of novel numerical methods for scientific and engineering applications. His doctoral work is focused on investigating and improving the properties of high-order generalized summation-by-parts discretizations in the context of problems of increasing practical relevance, for example, fluid flow problems governed by the steady and unsteady compressible Navier-Stokes equations. David received a B.Sc. degree in Mechanical Engineering from the University of Saskatchewan. He is currently a member of the Computational Aerodynamics Group at the University of Toronto Institute for Aerospace Studies under the supervision of Professor David W. Zingg.