Section: New Results

Cost reduction of numerical methods

This section gathers contributions for which the main motivation was to increase the efficiency of numerical methods, either by localizing the computational effort thanks to mesh refinement

In [24], E. Creusé and his collaborators generalize the equilibrated error estimators developed in the low-frequency magnetostatic case to the case of the harmonic time-dependent one. This contribution allows to obtain a bound of the numerical error equal to one, so that the accuracy of the obtained solution can be explicitly controlled.

The contribution [16] by K. Brenner and C. Cancès is devoted to the improvement of the behavior of Newton's method when solving degenerate parabolic equations. Such equations are very common for instance in the context of complex porous media flows. In [16], the presentation focuses on Richards equation modeling saturated/unsaturated flows in porous media. The basic idea is the following: Newton's method is not invariant by nonlinear change of variables. The choice of the primary variable then impacts the effective resolution of the nonlinear system provided by the scheme. The idea developed in [16] is then to construct an abstract primary variable to facilitate Newton's method's convergence. This leads to an impressive reduction of the computational cost, a better accuracy in the results and a strong robustness of the method w.r.t. the nonlinearities appearing in the continuous model.

In [39], Ward Melis, Thomas Rey, and Giovanni Samaey present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-€“Kutta method of arbitrary order. Based on the spectrum of the linearized BGK operator, they deduce that, with an appropriate choice of inner step size, the time step restriction on the outer time step as well as the number of inner time steps is independent of the stiffness of the BGK source term. They illustrate the method with numerical results in one and two spatial dimensions.

In [13], Christophe Besse, Guillaume Dujardin, and Ingrid Lacroix-Violet present the numerical integration in time of nonlinear Schrödinger equations with rotating term. After performing a change of unknown so that the rotation term disappears they consider exponential integrators such as exponential Runge-Kutta methods and Lawson methods. They provide an analysis of the order of convergence and some preservation properties of these methods and they present numerical experiments.