Nonlinear generalized functions and nonlinear numerical simulations in fluid and solid continuum mechanics

Abstract

We present numerical techniques based on generalized functions adapted to nonlinear calculations. They concern main numerical engineering problems ruled by-or issued from-nonlinear equations of continuum mechanics. The aim of this text is to invite the readers in applying these techniques in their own work without significant prerequisites by presenting their use on a sample of elementary applications from engineering. Pure mathematicians can read it easily since the numerical techniques are fully recalled in a very simple way and these applied mathematics are indeed based on nonlinear pure mathematics, namely on the use of generalized functions adapted to nonlinear calculations. The "nonlinear generalized functions" permit to compute explicit solutions of simple problems, called Riemann problems, in cases in which this is not possible using classical generalized functions (such as the distributions). These explicit solutions are the elementary building blocks of efficient numerical schemes needed by engineers.This text was prepared for an engineering meeting. For mathematicians it is an illustration of general introductions to these "nonlinear generalized functions". For physicists it complements, concerning continuum mechanics, the use of the nonlinear generalized functions in general relativity.