Adaptive time splitting method for multi-scale evolutionary partial differential equations

Description

This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady problems. The strategy considers a second order Strang method and another lower order embedded splitting scheme that takes into account potential loss of order due to the stiffness featured by time-space multi-scale phenomena. The scheme is then built upon a precise numerical analysis of the method and a complementary numerical procedure, conceived to overcome classical restrictions of adaptive time stepping schemes based on lower order embedded methods, whenever asymptotic estimates fail to predict the dynamics of the problem. The performance of the method in terms of control of integration errors is evaluated by numerical simulations of stiff propagating waves coming from nonlinear chemical dynamics models as well as highly multi-scale nanosecond repetitively pulsed gas discharges, which allow to illustrate the method capabilities to consistently describe a broad spectrum of time scales and different physical scenarios for consecutive discharge/post-discharge phases.

Abstract

Accepted to publication in Confluentes Mathematici. Dedication : Cet article est dédié à la mémoire de Michelle Schatzman. Spécialiste des méthodes de décomposition d'opérateur, sa grande clairvoyance scientifique lui a permis d'orienter plusieurs chercheurs débutants sur ce sujet à un moment où il pouvait sembler achevé. Michelle aimait dire qu'il n'y a pas de frontière entre les branches des mathématiques et que seule une grande culture permet de naviguer dans cette forêt et d'y trouver les bonnes techniques pour résoudre un problème. Ce travail est un hommage; à la croisée des mathématiques et de leurs applications effectives, il tente d'illustrer cette assertion. Michelle, ton dynamisme, ton humour et ton plaisir à parler mathématiques nous manquent.

Abstract

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