Abstract
In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.
Highlights
It is a well-known fact that transport equations do not require prescription of any boundary condition at an outflow boundary, that is, when the transport velocity is outgoing with respect to the boundary of the spatial domain
Such discretizations necessitate the prescription of numerical boundary conditions at an outflow boundary [GKO95, Str04], even though the underlying partial differential operator does not require any boundary condition for determining the solution
The approach in [Kre[66], Gol[77], GT81] is rather elaborate since it relies first on the verification of the so-called Uniform Kreiss-Lopatinskii Condition, and on the application of deep general results which show that the latter condition is sufficient for the derivation of optimal semigroup estimates
Summary
It is a well-known fact that transport equations do not require prescription of any boundary condition at an outflow boundary, that is, when the transport velocity is outgoing with respect to the boundary of the spatial domain. The approach in [Kre[66], Gol[77], GT81] is rather elaborate since it relies first on the verification of the so-called Uniform Kreiss-Lopatinskii Condition (that is, in the present context of numerical schemes, a refined version of the Godunov-Ryabenkii condition [GKS72, GKO95]), and on the application of deep general results which show that the latter condition is sufficient for the derivation of optimal semigroup estimates. Such general results first arose in [Kre66] and were later proved in further generality by Kreiss, Osher and followers [Kre[68], Osh[69], Wu95, CG11].
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