Jean-François Coulombel

Benjamin Boutin, Jean-François Coulombel

In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels, while semigroup estimates were restricted, up to now, to numerical schemes with two time levels only.

Jean-François Coulombel

The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems. We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss-Lopatinskii Condition introduced in [GKS72]. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.

Jean-François Coulombel

Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in Leray and Gårding's work provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu.

Jean-François Coulombel

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in â 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].