Special PDE Seminar
Scientific Computing Seminar
Abstract: Stability implies convergence, but for nonlinear problems we can do with less. To make our point, we focus on non-oscillatory {\it central schemes} as prototype for Godunov-type projection methods. A variety of numerical experiments demonstrate that the proposed central schemes offer simple, robust, Riemann-solver-free alternatives to the more expensive upwind schmes. Our new convergence results apply to general projection methods, including those which are based on modern nonlinear projections employed by both central as well as {\it high-resolution schemes}.
It is shown that if the approximate solution possess a weak regularity, then it converges to the unique entropy solution. The convergence result is obtained by interpolation between this weak regularity and well-posedness measured in an appropriate {\it negative norm}. Here, weak regularity is quantified in terms of a priori BV-bound, a weaker entropy-production bound, ... Most notably, since our theory does not require a stronger stability condition, one is able to apply these results to high-resolution approximations, and in this context we establish first convergence results of fully-discrete high-resolution approximations. In particular, we answer the long question concerning the convergence of the (fully-discrete) second-order MUSCL scheme and related high-resolution central approximations. These convergence results are complemented by the corresponding error estimates.
<--- 2000 Index