Richardson extrapolation is a numerical procedure by which we can eficiently enhance the accuracy of a time integration method. During this procedure a weighted average of two numerical solutions obtained by the same numerical method, but two different values of the time-step sizes is calculated. In this talk we present the theoretical background of Richardson extrapolation, and give some results about its convergence and stability. The applicability of the method will be illustrated on different environmental models (simlplified carbon-dioxide model, air chemistry model, advection equation). (The talk will be in Hungarian.)
A szeminárium honlapja: http://math.bme.hu/alkanalszemi
Abstract: Space and time discretisations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual inner product is replaced by an suitable inner product on a problem specific Hilbert space. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semi-linear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretisation by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration, which yields surprising numerical results.