Abstract: We consider a class of semi Lagrangian schemes for linear and fully non-linear diffusion equation. The schemes, which are a type of easy to implement difference+interpolation wide stencil schemes, includes and generalize previous schemes of Menaldi, Camilli–Falcone and (in some cases) Crandall–Lions. The schemes are by construction always monotone and hence of low order (typically 1st order). Their main advantage is that they are consistent (and stable) for ANY (smooth) positive semidefinite diffusion matrix. The last property is important in some applications with degenerate diffusions, and it is not shared by conventional schemes like e.g. finite difference schemes. For finite difference schemes the best you can do seems to be to approximate the diffusion matrix by “more diagonal dominant” matrices and use ever wider stencils in its approximation, cf. Bonnans–Zidani. This, in general rather complicated approximation proceedure, is avoided by the semi Lagrangian schemes. We will give some background, discuss construction, monotonicity, stability, convergence, error estimates, and present simulations. We will also discuss some cases where a 2nd order compact stencil scheme can be achieved.

Abstract: We consider the KdV equation u_t+(u²)_x+u_xxx=0 used for modeling, e.g., water waves in a narrow channel. The time evolution of this nonlinear partial differential equation is determined by a quadratic (Burgers') term (u²)_x plus a linear (Airy) term u_xxx. Since the evolution of each of these terms is quite different, it is natural to ask whether the concatenation of the evolution operators for each term, i.e., u_t+(u²)_x=0 and u_t+u_xxx=0$, yields an approximation to the evolution operator for the sum. This strategy has been used with good results when designing numerical methods for the KdV equation, but without rigorous convergence proofs. The aim of this talk is to present a first step in this direction, and show convergence of operator splitting for sufficiently regular initial data. (Joint work with N. H. Risebro, K. H. Karlsen, T. Tao.)

Abstract: One of the most famous examples of comletely integrable wave equations is the Korteweg–de Vries equation, q_t(x,t)=6q(x,t)q_x(x,t)−q_xxx(x,t), which can be solved via the inverse scattering method. We will consider decaying initial data and show how the inverse scattering problem can be reformulated as a Riemann–Hilbert problem. Based on this we will present how to extract long-time asymptotics by using the nonlinear steepest descent method.