Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

Abstract

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.

Discrete transparent boundary conditions for the two-dimensional leap-frog scheme: approximation and fast implementation

“…As in all numerical approximation problems, a crucial feature for the efficiency of the numerical scheme is stability. Following the general result of [15], the DTBC for the leap-frog scheme on a half-space do not meet the strongest possible stability properties because the leap-frog scheme exhibits glancing wave packets (even in one space dimension). On a half-space, the exact DTBC meet some kind of neutral stability which calls for special care since a slight modification in the numerical boundary conditions may yield violent instabilities.…”

“…For later use and also for the ease of reading, we first go back to the corresponding one-dimensional problem in Section 3. We adapt the general strategy of [3,18], which was recently generalized in [15], to the particular case of the leap-frog scheme. Some of the tools used in [15] are complemented here by some explicit expansions, based on Legendre polynomials, which were already used in [3,12,18].…”

“…We adapt the general strategy of [3,18], which was recently generalized in [15], to the particular case of the leap-frog scheme. Some of the tools used in [15] are complemented here by some explicit expansions, based on Legendre polynomials, which were already used in [3,12,18]. Such explicit representation formulae lower the implementation cost of our method and give a sharp asymptotic behavior for the convolution coefficients, which would not be possible by only using the tools in [15].…”

“…Some of the tools used in [15] are complemented here by some explicit expansions, based on Legendre polynomials, which were already used in [3,12,18]. Such explicit representation formulae lower the implementation cost of our method and give a sharp asymptotic behavior for the convolution coefficients, which would not be possible by only using the tools in [15]. We then analyze the stability of the one-dimensional DTBC by means of the normal mode decomposition and show that they satisfy the so-called Godunov-Ryabenkii condition [22].…”

“…By "analytically available", we mean that DTBC can be expressed by a welldefined operator, even though its practical implementation may be difficult. We refer for instance to [2][3][4]15] and references therein for various aspects of the theory, the main features of which are recalled below. In either case, the discrete spatial domain is a cylinder of the form × N with no tangential boundary (the variables in ) so DTBC can be derived by performing either a Fourier transform or a Fourier series decomposition with respect to the spatial tangential variables.…”