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Nonlinear edge diffusion and the discrete maximum principle

Abstract:. Nonlinear edge diffusion and the discrete maximum principle In this talk I will review recent results on the analysis of shock-capturing-type methods applied to convection-dominated problems. The method of choice is a variant of the Algebraic Flux-Correction (AFC) scheme. This scheme has received some attention over the last two decades due to its very satisfactory numerical performance. Despite this attention, until very recently there was no stability and convergence analysis for it. Thus, the purpose of the works reviewed in this talk was to bridge that gap. The first step towards the full analysis of the method is a rewriting of it as a nonlinear edge-based diffusion method. This writing makes it possible to present a unified analysis of the different variants of it. So, minimal assumptions on the components of the method are stated in such a way that the resulting scheme satisfies the Discrete Maximum Principle (DMP) and is convergent. One property that will be discussed in detail is the linearity preservation. This property has been linked to the good performance of methods of this kind. We will discuss in detail its role and the impact of it in the overall convergence of the method. Time permitting, some results on a posteriori error estimation will also be presented.

This talk will gather contributions with A. Allendes (UTFSM, Chile), E. Burman (UCL, UK), V. John (WIAS, Berlin), F. Karakatsani (Chester, UK), P. Knobloch (Prague, Czech Republic), and R. Rankin (U. of Nottingham, China).

This seminar will take place on Friday 27th April, at 4p.m. in Room A2-002

If you have any questions regarding this seminar, please direct them to Iain Moyles (061 233726,

A full list of upcoming seminars can be found at

Supported by Science Foundation Ireland funding, MACSI - the Mathematics Applications Consortium for Science and Industry (, centred at the University of Limerick, is dedicated to the mathematical modelling and solution of problems which arise in science, engineering and industry in Ireland.