Approximated lax pairs and empirical interpolation for nonlinear parabolic partial differential equations

Jean-Frédéric Gerbeau 1 Damiano Lombardi 1 Elisa Schenone 2, 1
1 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, UPMC - Université Pierre et Marie Curie - Paris 6, Inria de Paris
Abstract : 1. Introduction. Several methods exist to approximate the solutions of PDEs with a reduced number of degrees of freedom (Reduced Basis Methods, Proper Orthogonal Decomposition, etc.). But time-dependent problems dominated by transport or propagation phenomena remain a challenge for reduced-order modelling. In [6], a new approach was proposed to approximate the nonlinear evolution partial differential equations. This method, called ALP, is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, it is not based on an off-line / on-line strategy, and the basis on which the solution is searched for evolves in time according to a dynamics specific to the problem. This algorithm was successfully used for various nonlinear problems, including the equations of cardiac electrophysiology [7], which exhibit front propagation. Two weaknesses of the original method were the limitation to polynomial nonlinearities and a complexity proportional to the power three of the number of modes. In this article, a new algorithm is proposed which does not suffer from these two shortcomings. The ALP-EI method introduced in this work combines the idea of ALP, i.e. a time evolving basis which does not rely on a database, and the treatment of nonlinear terms using a selection of spatial points from an Empirical Interpolation technique in the same spirit as in [1, 3]. More specifically, the definition of the basis functions and of their time evolution does not change compared to the ALP method, but the functions and the operator used to solve the reduced order problem are evaluated on a finite set of points which is typically a small subset of the finite element nodes. As a consequence, the ALP-EI method is well-suited to non-polynomial nonlinearities, and much less expensive than the original ALP method.
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Jean-Frédéric Gerbeau, Damiano Lombardi, Elisa Schenone. Approximated lax pairs and empirical interpolation for nonlinear parabolic partial differential equations . 2016. ⟨hal-01278778⟩

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