Stéphane Descombes

Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation

In this paper, we consider the nonlinear Schrodinger equation

Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction

u_t+i\Delta u -F(u)=0

Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations

in two dimensions. We show, by an operator-theoretic proof, that the well-known Lie and Strang formulae (which are splitting methods) are approximations of the exact solution of order 1 and 2 in time.

Structure of Korteweg models and stability of diffuse interfaces

Summary.In this paper, we perform the numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term. We consider both linear diagonal and quasi-linear non-diagonal diffusion; the entropic structure implies the well-posedness and stability of the system as well as a Tikhonov normal form for the nonlinear reaction term [23]. It allows to perform a singular perturbation analysis and to obtain a reduced and well-posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution. We then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step Δt thus leading to the failure of the usual splitting analysis techniques. We define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales. All the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level. We finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis.

Convergence of a splitting method of high order for reaction-diffusion systems

This paper addresses the construction of nonlinear integro-differential artificial boundary conditions for one-dimensional nonlinear cubic Schrodinger equations. Several ways of designing such conditions are provided and a theoretical classification of their accuracy is given. Semidiscrete time schemes based on the method developed by Duran and Sanz-Serna [IMA J. Numer. Anal. 20 (2000), pp. 235-261] are derived for these unusual boundary conditions. Stability results are stated and several numerical tests are performed to analyze the capacity of the proposed approach.

New Resolution Strategy for Multiscale Reaction Waves using Time Operator Splitting, Space Adaptive Multiresolution, and Dedicated High Order Implicit/Explicit Time Integrators

The models considered are supposed to govern the motion of compressible fluids such as liquid-vapor mixtures endowed with a variable internal capillarity. Several formulations and simplifica- tions are discussed, from the full multi-dimensional equations for non-isothermal motions in Eulerian coordinates to the one-dimensional equations for isothermal motions in Lagrangian coordinates. Hamil- tonian structures are pointed out in each case, and in the one-dimensional isothermal case, they are used to study the stability of two kinds of non-linear waves: the solitary, or homoclinic waves, and the het- eroclinic waves, which correspond to propagating phase boundaries of non-zero thickness, also called diffuse interfaces. It is known from an earlier work by Benzoni-Gavage (Physica D, 2001) that the lat- ter are (weakly) spectrally stable. Here, diffuse interfaces are shown to be orbitally stable. The proof relies on their interpretation as critical points of the Hamiltonian under constraints, whose justification requires some care because of the different endstates at infinity. Another difficulty comes from higher order derivatives that are not controlled by the Hamiltonian. In the case of a variable capillarity, our stability result unfortunately does not imply global existence. As regards the solitary waves, which come into families parametrized by the wave speed, they are not stable from the variational point of view. However, using a method due to Grillakis, Shatah and Strauss (Journal of Functional Analysis, 1987), it is possible to show that some solitary waves, depending on their speed, are orbitally stable. Namely, the convexity of a function of the wave speed called moment of instability determines the stability of soli- tary waves. This approach, already used by Bona and Sachs (Communications in Mathematical Physics, 1988) for the Boussinesq equation, is here adapted to solitary waves in Korteweg models, which are first classified according to their endstate and internal structure. The corresponding moments of instability are computed by quadrature. They exhibit both convexity and concavity regions. AMS classification. 76T10; 35B35; 35Q51; 37C29.

Strang's formula for holomorphic semi-groups

In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form u t - MΔu + F(u) = 0 where M is an m × m matrix whose spectrum is included in {Rz > 0}. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell's Equations

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. The paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two-dimensional spiral waves and three-dimensional scroll waves, as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed.

A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

Let E(t)=exp(−t(A+B)) and let W(t) be the Strang approximation to E(t): W(t)=exp(−tA/2)exp(−tB)exp(−tA/2). In this article, we give a formal Taylor expansion with remainder for W(t), where the derivation operator is replaced by the bracket with one of the operators A or B. We validate this expansion in several functional cases where A and B generate a holomorphic semi-group. They include the case of the Kac transfer operator, and the case A=−MΔ with M a non-necessarily diagonal matrix with spectrum included in {Rz>0} and B the multiplication by a spatially dependent matrix. We infer stability estimates and estimates on ∥E(t)−W(t/n)n∥ when n tends to infinity.

ADAPTIVE TIME SPLITTING METHOD FOR MULTI-SCALE EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.