Numerical solution of partial differential equations in time-dependent domains

Numerical solution of heat transfer and fluid flow problems in two spatial dimensions is studied. An arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations is applied to handle time-dependent geometries. A Legendre spectral method is used for the spatial discretization, and the temporal discretization is done with a semi-implicit multi-step method. The Stefan problem, a convection-diffusion boundary value problem modeling phase transition, makes for some interesting model problems. One problem is solved numerically to obtain first, second and third order convergence in time, and another is used to illustrate the difficulties that may arise with distribution of computational grid points in moving boundary problems. Strategies to maintain a favorable grid configuration for some particular geometries are presented. The Navier-Stokes equations are more complex and introduce new challenges not encountered in the convection-diffusion problems. They are studied in detail by considering different simplifications. Some numerical examples in static domains are presented to verify exponential convergence in space and second order convergence in time. A preconditioning technique for the unsteady Stokes problem with Dirichlet boundary conditions is presented and tested numerically. Free surface conditions are then introduced and studied numerically in a model of a droplet. The fluid is modeled first as Stokes flow, then Navier-Stokes flow, and the difference in the models is clearly visible in the numerical results. Finally, an interesting problem with non-constant surface tension is studied numerically.