L-stable implicit trapezoidal-like integrators for the solution of parabolic partial differential equations on manifolds

A new Trapezoidal-type scheme is proposed for the direct numerical integration of time-dependent partial differential equations. The evolving system of ordinary differential equations after discretization is usually stiff, so it is desirable for the use of a numerical method in solving it to have good properties concerning stability. The method proposed in this article is L-stable and at least of algebraic order two. It is increasingly common to encounter partial differential equations (PDEs) posed on manifolds, and standard numerical methods are not available for such novel situations. Here, an L-Stable implicit Trapezoidal-like numerical integrator was developed for solving partial differential equations on manifolds. This approach allows the immediate use of familiar finite difference methods for the discretization and numerical solution. Presented here are the motivation and details of the method, illustration of its numerical convergence and stability properties for a general case. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable of some PDE problems. Results show accuracy of maximum error for various space and time steps.   Key words: Trapezoidal-type method, partial differential equations, L-stability, implicit surfaces, manifolds, stiff systems.