Galerkin/Runge-Kutta discretizations for parabolic partial differential equations

Efficient, high order Galerkin/Runge-Kutta methods are constructed and analyzed for certain classes of parabolic initial boundary value problems. In particular, the partial differential equations considered are (1) semilinear, (2) linear with time dependent coefficients, and (3) quasilinear. Optimal order error estimates are established for each case. Also, for the problems in which the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximatley. Nevertheless, the resulting algorithm is shown to preserve the optimal order convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.