An oscillating plate means the code has to be suited for arbitrary geometries and for moving domain boundaries. We employ a Godunov-type finite-volume scheme with the Osher/Solomon approximate Riemann fluxes. A general modification extends Riemann solvers to moving cell boundaries. Occasionally, grid distortion might require switching to a new, regular grid (locally or everywhere); to this end we propose a conservative "rezoning" algorithm. To increase the spatial accuracy, a high-resolution scheme with a new kind of slope limiter for arbitrary grid topologies is developed. Furthermore, we discuss various choices of time integration. Implicit schemes require solving a nonlinear equation with Newton iteration and GMRES. GMRES convergence is accelerated with standard block Jacobi and Gauß-Seidel preconditioners; we also propose a choice of index ordering for the latter.
We present numerical experiments for a range of simple and complicated test problems to demonstrate the quality (or lack thereof) of various methods. Our code solves the 2D Navier-Stokes equations; many of the techniques we describe can easily be generalized to any number of dimensions.
Finally, we briefly discuss the problem of choosing large time steps in presence of discontinuities by having individual time steps for different parts of the domain. We propose a highly adaptive 1D algorithm and examine the savings for linear advection and Burgers equation.