Stable High-Order Finite Difference Methods for Aerodynamics

University dissertation from Uppsala : Acta Universitatis Upsaliensis

Abstract: In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates.The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version.With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes.We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

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