Operator splitting methods with error estimator and adaptive time-stepping. Application to the simulation of combustion phenomena

Operator splitting techniques were originally introduced with the main objective of saving computational costs. A multi–physics problem is thus split in subproblems of different nature with a significant reduction of the algorithmic complexity and computational requirements of the numerical solvers. Nevertheless, splitting errors are introduced in the numerical approximations due to the separate evolution of the split subproblems and can compromise a reliable reproduction of the coupled dynamics. In this chapter we present a numerical technique to estimate such splitting errors on the fly and dynamically adapt the splitting time steps according to a user–defined accuracy tolerance. The method applies to the numerical solution of time–dependent stiff PDEs, illustrated here by propagating laminar flames investigated in combustion applications. 1 Context and motivation Let us consider a scalar reaction–diffusion equation ∂tu− ∂ xu = f(u), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, } (1) and represent the solution u(., t) as T u0, where T t is the semi–flow associated with (1). Given v0 and w0, an operator splitting approach amounts to consider the following subproblems: ∂tv − ∂ xv = 0, x ∈ R, t > 0, v(x, 0) = v0(x), x ∈ R, } (2) and ∂tw = f(w), x ∈ R, t > 0, w(x, 0) = w0(x), x ∈ R. } (3) ∗Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France and INRIA Sophia Antipolis-Mediterrané Research Center, Nachos project-team, 06902 Sophia Antipolis Cedex, France ([email protected]). †CCSE, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. MS 50A-1148, Berkeley, CA 94720, USA and CD-adapco, 200 Shepherds Bush Road, London W6 7NL, UK ([email protected]). ‡CNRS UPR 288, Laboratoire EM2C, CentraleSupélec, Fédération de Mathématiques de l’École Centrale Paris, CNRS FR 3487, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France ([email protected]).