Konvergenz Finiter Differenzenverfahren für nichtlineare hyperbolisch-parabolische Systeme

Sammanfattning: This thesis presents a new technique to prove the convergence of finite difference methods applied to nonlinear Systems arising in computational fluid dynamics.

The underlying Systems are either hyperbolic such as the Euler equations or mixed hyperbolic-parabolic like the Navier-Stokes equations. We analyze implicit finite-difference methods. As the argument is based on the concept of consistency and stability, we obtain convergence results for classical Solutions. An important point is that the results are achieved unconditional to the order of consistency of the scheme. It is known that the construction of high-order methods in computational fluid dynamics is much more difficult than in the case of ordinarydifferential equations.

The analysis uses ideas from Löpez-Marcos and Sanz-Serna [16] to establish local stability regions around a pilot function. The introduction of this pilot function is the essential idea leading to success. The pilot function is constructed in such a way that it is highly consistent with the scheme and that it converges to the Solution of the underlying system when the stepsize tends to zero.

Strang made use of a similar idea [22]. But, in contrast to Strang's
concept, we do not use the stability of the scheme linearized at the
classical Solution of the partial differential equation. Instead we linearize the scheme at the pilot function. By means of energy estimates, we show the stability of this linearization. Finally, the convergence of the Solution of the discrete method to the pilot function as well as to the desired classical Solution is proved.