The closest point method for time-dependent processes on surfaces

This thesis concerns the numerical solution of time-dependent partial differential equations (PDEs) on general surfaces using a recent technique known as the Closest Point Method. The Closest Point Method represents surfaces with a closest point representation which leads to great flexibility with respect to surface geometry, among other advantages. The computation itself alternates between two steps: first, an explicit time step is performed using standard finite difference techniques on a narrow band of grid points surrounding the surface embedded in a higher dimension; second, a closest point extension is used to maintain consistency with the original surface PDE. The Closest Point Method is applied to the important problem of interface motion on surfaces by using level set equations posed on surfaces. New weighted essentially non-oscillatory (WENO) interpolation schemes are derived to perform the necessary closest point extensions. This approach, in combination with standard Hamilton--Jacobi WENO finite difference schemes and explicit time stepping, gives high-order results (up to fifth-order) on a variety of test problems. Example computations are performed on a sphere, torus, triangulated human hand and Klein bottle to demonstrate the flexibility of the method. A new implicit Closest Point Method is presented for surface PDEs which are stiff, for example, because of diffusion terms. The method uses implicit time-stepping schemes to allow large steps but retains the flexibility with respect to surface geometry of the original explicit Closest Point Method. Numerical convergence studies on the heat equation and a fourth-order biharmonic problem demonstrate the accuracy of the method and a variety of example computations demonstrate its effectiveness. These include an image processing example of blurring on triangulated surfaces, heat diffusion on a surface consisting of multiple components connected by a thin filament and Turing pattern formation on surfaces using implicit--explicit (IMEX) time stepping. A class of time-stepping methods known as diagonally split Runge--Kutta (DSRK) methods is investigated. These methods appear promising because they offer both high-order convergence and unconditional contractivity (a nonlinear stability property). However, numerical computations and analysis of stage-order demonstrates that unconditionally contractive DSRK methods suffer from order reduction which severely limits their practical application.