On maximum norm contractivity of second order damped single step methods

In this paper we consider A(θ)-stable finite difference methods for numerical solutions of dissipative partial differential equations of parabolic type. Combining two rational approximation methods with different orders of accuracy, where the lower order method is applied n0 times (n0 fixed) at each time step, we prove the existence of a second order method which is contractive for all time steps. Moreover, we shed light on the conditions on the lower order method which are sufficient (and sometimes necessary) to obtain the optimal order of accuracy. For the one-dimensional heat equation we construct a family of numerical methods which are contractive in the maximum norm for all values of the discretization parameters. We also present numerical examples to illustrate our results.