Runge-Kutta time discretization of nonlinear parabolic evolution equations.

Research project P 13754 Time discretization of parabolic evolution equations Alexander OSTERMANN 11.10.1999 In recent years nonlinear evolution equations of parabolic type have gained a lot of interest, since they are increasingly used to describe phenomena that arise in various fields of applications. As examples we mention the incompressible Navier-Stokes equations from fluid dynamics, the Bellman equations from stochastic control, the nonlinear Cahn-Hilliard equation from pattern formation in phase transitions, and reaction-diffusion equations that are used to model air pollution. A key problem in the numerical solution of such equations is to determine how well the dynamics of the underlying initial value problem is captured by the discretization. The answer to this question helps to interprete correctly the numbers obtained from a computer simulation. Within the past years, the knowledge on stability and error bounds for discretizations of linear and semilinear problems has grown considerably. For the nonlinear case, though, only few convergence results are known. The aim of the present project is to obtain convergence estimates for Runge-Kutta time discretizations of nonlinear parabolic equations. A new analytical framework is used that allows to apply ideas from the semilinear case. In a first step, error estimates for smooth solutions will be derived. They characterize the behaviour of the numerical approximation for short-time integration. These convergence bounds, however, tend to break down in the case of very long times, since the involved constants may depend exponentially on the length of the time interval. In recent years, it has become more and more evident that error bounds for non-smooth initial data on bounded time intervals are a basic ingredient in the study of the long-term behaviour of numerical methods. Therefore, we also plan to derive non-smooth data error estimates for nonlinear parabolic problems. With these error estimates at hand, it is possible to study the long-term behaviour of numerical discretizations. For instance, one could investigate the approximation of sets that are invariant under the flow of the evolution. We plan to study these possibilities.

Nonlinear evolution equations of parabolic type play an important role in diffusion processes. They are used in the modeling of pattern formation in phase transitions as well as in air pollution models and in the field of medical image processing. Within the scope of our project we could show that strongly stable Runge-Kutta methods are able to solve the class of fully nonlinear parabolic equations in an efficient and reliable way. These results form the basis of computer programs and allow in certain situations to interpret correctly the numbers obtained from computer simulations. The main goal of our project was to analyze the behavior of numerical solutions of fully nonlinear parabolic problems. This class of partial differential equations is frequently used in modeling nonlinear diffusion processes and has thus wide applications. If the nonlinearity is small with respect to the diffusion (and the diffusion thus dominates in some sense the dynamics), it is well known how numerical methods behave. We treated within our project the much more difficult problem of large nonlinearities that typically appear in fully nonlinear equations. Large nonlinearities require totally new concepts. We took advantage of a quite new analytical framework by A. Lunardi which could be adapted to our situation and which simplified a lot some previous attempts. The essential point was that we could establish a (generalized) variation of constants formula which turned out to be the basis of our analysis. Our main goal was to study the behavior of Runge-Kutta approximations. Due to their excellent stability properties, implicit Runge-Kutta methods are the ideal candidates for time integrators in our situation. After proving the existence of numerical solutions, we were able to prove convergence results and thus obtained bounds for the errors of the numerical approximation. Such error bounds are essential since they form the theoretical basis for computer programs. Further we investigated the qualitative behavior of Runge-Kutta approximations. The key problem there is to determine how well the underlying dynamics of the problem is captured by the numerical discretization. In the case of hyperbolic equilibrium points, we have found a positive answer: our numerical discretization has the same dynamics as the underlying analytical problem. We published or results in renowned journals (Mathematics of Computation, Applied Numerical Mathematics) and we further presented them at various international conferences and colloquia.