Efficient Fully Discrete Summation-by-parts Schemes for Unsteady Flow Problems

We make an initial investigation into the numerical efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier-Stokes equations we consider a two-dimensional advectiondiffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summationby-parts operators, and compare the results to other popular high order methods. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.