Abstract: High-order finite difference schemes are notoriously difficult to stabilize at boundaries. A successful remedy for this problem is to use Summation-by-Parts (SBP) finite difference schemes along with Simultaneous Approximation Terms (SAT) for imposing boundary conditions. I will review this theory and indicate how linear stability can be proved via energy estimates for the full Navier-Stokes equations on multi-block grids. Linear stability implies that a smooth solution remains bounded under small perturbations. For subsonic flows this generally guarantees stability but for transonic and supersonic flows the solution may develop discontinuities and the linear theory breaks down. I will discuss the extension of the SBP theory to flows with non-smooth solutions. For such flows, entropy plays the same role as energy for linear problems and is the key to stable approximations. I will show how entropy stable high-order finite difference schemes can be constructed for Cauchy problems and also discuss problems that arise when boundaries are introduced.
Tuesday, April 27, 2010