A high order splitting method for time-dependent domains

preprint numerics no. 1/2008 norwegian university of science and technology trondheim, norway We present a temporal splitting scheme for the semi-discrete convection-diffusion equation and the semi-discrete incompressible Navier-Stokes equations in time-depedent geometries. The proposed splitting scheme can be considered as an extension of the OIF-method proposed in [22] in the sense that it can be interpreted as a semi-Lagrangian method for time-dependent domains. The semi-discrete equations are derived from an arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations, and are discretized in space using high order spectral elements. The proposed splitting scheme has been tested numerically on model problems with known analytical solutions, and first, second , and third order convergence in time has been obtained. We also show that it is not necessary for the interior mesh velocity to be obtained through the use of an elliptic solver. Numerical tests show that it is sufficient that the mesh velocity is regular within each spectral element and only C 0-continuous across element boundaries; this is consistent with the theoretical results presented in [9]. In addition, the mesh velocity should be regular in the time direction.