Numerical and asymptotic studies of complex flow dynamics. Progress report, January 1--December 31, 1997

If one has computed a stationary state of an evolution equation, a major question is if the state is stable, and many different techniques have been developed to discuss this important question. Together with H.-O. Kreiss the author completed a survey article on nonlinear stability for time dependent PDEs. They give an account of the Lyapunov technique and the resolvent technique to study stability questions. It has been known for some time that the strength of the resolvent technique is the control of the small-wave-number projection of the solution, whereas Lyapunov`s technique is good for high wave numbers. In ref. 6 they succeeded in combining the two techniques. Together with T. Hagstrom, the author has studied somewhat related questions for flows at low Mach number. They show all time existence of classical solutions when the initial data are almost incompressible. The result is stronger than Hoff`s in that they can allow for a ball of slightly compressible data with a radius independent of the Mach number M. In contrast, the radius of Hoff`s ball shrinks to zero as M approaches 0. Together with H.-J. Schroll, RWTH Aachen, the author worked on conservation laws with stiff source term. From a mathematical point of view, these are singular perturbation problems for which the reduced problem is singular. In ref. 2 they introduce the notion of stiff well-posedness, which--in the linear constant coefficient case--characterizes all hyperbolic systems with stiff source term 1/{epsilon} Bu whose solutions converge for {epsilon} approaches 0. There are many applications. A rather new field, which has received considerable attention in the physics community, is self-organized criticality. Bak gives a popular account. The aim is to study the self-organization of systems towards a critical state and to understand the scaling properties at criticality. Together with the students the author has analyzed a model and has performed numerical simulations