Half-time seminar - David Niemelä: "Summation-by-parts finite differences for fluid dynamic equations, solving incompressible and compressible fluid flows"

Welcome to my half-time seminar: "Summation-by-parts finite differences for fluid dynamic equations, solving incompressible and compressible fluid flows"

External Reviewer: Jan Glaubitz (Linköping University).

Abstract: In this seminar I will present the work from three manuscripts concering the development of high-order, stable, and accurate numerical methods for fluid dynamics problems using summation-by-parts (SBP) operators.

First, I discuss the work on a stable and accurate high-order solver for the incompressible Navier-Stokes equations using SBP operators on multi-block curvilinear grids. We apply the projection method to connect the interior multi-block interfaces, and show the stability of the discretization through a discrete version of the energy method. Analytical test cases confirm that the discretization achieves good accuracy and high-order convergence. Additionally, our method shows excellent agreement with benchmark data for a well-studied benchmark problem.

Second, I present work on the Euler and compressible Navier-Stokes equations applying a high-order artificial viscosity method, Residual Viscosity, using upwind SBP operators. The scheme shows robust and accurate solutions for under-resolved flows and in shock-dominated regimes. Benchmark problems such as the Sod shocktube, a Riemann problem, and the viscous shock tube problem of Daru and Tenaud, demonstrate accurate shock-capturing and expected convergence behaviour.

Finally, the work on optimised upwind SBP operators. By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm SBP framework, ensuring linear stability on poecewise curvilinear multiblock grids. The proposed oeprators yield significantly improved accuracy and computational efficiency compared with SBP operators constructed on equispaced grids. The accuracy and stability properties of the proposed operators are demonstrated through numerical experiments for linear hyperbolic problems in one spatial dimension and for the compressible Euler equations in two spatial dimensions.

The seminar will be held in english. Welcome!