Geometric numerical integration and Schrödinger equations.
Erwan Faou
European Math. Soc. To appear

Chapter I: Introduction

In this Chapter, we present the nonlinear Schrödinger equations (NLS) mainly considered in this lecture and describe the splitting algorithms used to approximate its solutions. Numerical experiments are presented to show that the preservation of the energy and the correct reproduction of energy exchanges between the modes are not automatically guaranteed by the numerical solution over long time. Clues are given to explain this phenomenon: it relies on the use of CFL conditions, on possible numerical resonances induced by a bad choice of the stepsize, or even on the arithmetic nature of the number of grid points used in the simulations. These numerical experiments are done in different cases: the simulation of solitary waves for NLS, the linear situation, the case of a small nonlinear perturbation of the resonant nonlinear Schrödinger equation. The goal of this lecture will be to give mathematical explanations of these numerical observations.

Chapter II: Finite dimensional backward error analysis

Before tackling the infinite dimensional case (PDE), the goal of this chapter is to show that in a finite dimensional situation, the splitting schemes described in the previous chapter always (almost) preserve the energy. This result is given by the backward error analysis theory developped in the nineties to explain the good behavior of symplectic integrators applied to Hamiltonian systems of ordinary differential equations (ODE). The proof relies on the Baker-Campbell-Hausdorff formula, and the result is that the numerical solution produced by splitting methods can be interpretated as the exact solution of a modified Hamiltonian system over very long time. As this modified Hamiltonian system is close to the original one, this ensures the preservation of the energy of the numerical solution over very long time.

Chapter III: Infinite dimensional and semi-discrete Hamiltonian flows.

We now discuss the existence and uniqueness of solutions of the nonlinear Schrödinger equation. To this aims, we introduce functions spaces that are very convenient to handle polynomial nonlinearities. Theses spaces are based on the Wiener algebra defined as the set of functions whose Fourier transform is integrable. In such spaces, we will show the existence and uniqueness of local solutions to NLS, and discuss the global existence in some specific situations. We focus on the Hamiltonian structure of NLS and discuss the properties of the associated infinite dimensional flow. We end this Chater by considering semi-discrete Hamiltonian flows obtained by space discretization with pseudo-spectral Fourier collocation methods.

Chapter IV: Convergence results

We consider splitting methods applied to semilinear Schrödinger equations with a polynomial nonlinearity. We give some convergence results in the following sense: if the solution is smooth enough, then the splitting methods approximates the exact solution up to some convergence rate depending on the time step. We show that the Lie splitting method is of local order 2 and global order 1 over finite time intervals. We then consider implicit explicit integrators, or more generally smoothed schemes where the high frequencies of the linear part are tempered. Finally, we give similar result in the fully discrete case, where the solution is discretized both in space and time.

Chapter V: Modified energy in the linear case

We consider the case of the linear Schrödinger equation on the d-dimensional torus with a linear smooth potential. We show that for implicit-explicit integrators and for splitting methods with CFL condition, there exists a modified energy which is exactly preserved along the numerical flow. We then consider the case of fully discrete solution (in space and time), and show the existence of uniform bounds for the numerical solution over all time, provided the initial solution is smooth enough.

Chapter VI: Modified energy in the semilinear case

In the continuation of the previous Chapter, we prove backward error analysis results for splitting methods applied to the nonlinear Schrödinger equation with cubic nonlinearity. We show that the existence of a modified polynomial energy is guaranteed on a time depending in general on a CFL condition imposed on the stepsize, and is valid as long as the solution remains in a bounded set of the Wiener albegra. We then extend this result to the case of fully discrete numerical solutions. In particular, we prove that for sufficiently small initial data, the fully discrete numerical approximation NLS has almost global existence with a time depending only on the CFL condition imposed to the system.

Chapter VII: Introduction to long time analysis

In this final Chapter, we consider the resonant nonlinear Schrödinger equation with cubic nonlinearity. The term resonant is used in contrast to the situation where the linear Laplace operator is perturbed by a potential (whose frequencies are all integers). Moreover, we consider only small initial data, or equivalently small nonlinearity in the initial equation. Using a resonance analysis, we show that the situation differs significantly between the dimensions 1 or 2. In dimension 1, we can prove the preservation of the actions over long time, while in the same time scale, it is possible to prove the existence of an energy cascade from low to high modes in dimension 2. Using the results of the previous Chapter, we then discuss the persistence of these qualitative behaviors to numerical approximations. We show that it relies in general on the CFL number, the arithmetic nature of the number of grid points, or the properties of the numerical integrator (implicit or explicit).