A high order splitting method for time-dependent domains

We present a temporal splitting scheme for the semi-discrete convection-diffusion equation and the semi-discrete incompressible Navier-Stokes equations in time-dependent geometries. The proposed splitting scheme can be considered as an extension of the OIF-method proposed in Maday et al. [Y. Maday, A.T. Patera, E.M. Ronquist, An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow. J. Sci. Comput. 5 (4) (1990) 263-292] 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 e-continuous across element boundaries; this is consistent with the theoretical results presented in Formaggia and Nobile [L. Formaggia, F. Nobile, A stability analysis for the arbitrary Lagrangian-Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (2) (1999) 105-131]. In addition, the mesh velocity should be regular in the time direction. (c) 2008 Elsevier B.V. All rights reserved.