Amplification of numerical wave packets for transport equations with two boundaries

TL;DR

This work analyzes the stability of a fixed-parameter finite-difference discretization of the 1D transport equation on an interval when Dirichlet data are imposed at the inflow and Neumann extrapolation is used at the outflow. By combining von Neumann stability with wave-packet and boundary-reflection arguments, it demonstrates that stable interior schemes can exhibit large-time exponential amplification due to boundary coupling, depending on the stencil and extrapolation order. The authors provide explicit high-order stencil examples (third- and second-order extrapolation) that exhibit amplification consistent with the spectral radius $\rho(A)$, highlighting that commonly cited stability criteria may fail in the interval setting. The findings stress the need for careful boundary treatment design and verification for high-order schemes to avoid hidden instabilities in practical computations.

Abstract

The purpose of this note is to investigate the coupling of Dirichlet and Neumann numerical boundary conditions for the transport equation set on an interval. When one starts with a stable finite difference scheme on the lattice $\mathbb{Z}$ and each numerical boundary condition is taken separately with the Neumann extrapolation condition at the outflow boundary, the corresponding numerical semigroup on a half-line is known to be bounded. It is also known that the coupling of such numerical boundary conditions on a compact interval yields a stable approximation, even though large time exponentially growing modes may occur. We review the different stability estimates associated with these numerical boundary conditions and give explicit examples of such exponential growth phenomena for finite difference schemes with ''small'' stencils. This provides numerical evidence for the optimality of some stability estimates on the interval.

Figures (5)

• Figure 1: Left: amplification factor (dark red curve) $\theta \mapsto \sum_{\ell=-r}^p \, a_\ell \, {\rm e}^{\mathbf{i} \, \ell \, \theta}$ with $r=p=7$ and coefficients $a_\ell$ given in \ref{['coeff1']} within the unit circle $\mathbb{S}^1$ (blue curve). Right: zoom of the amplification factor near the tangency point at $z=1$.
• Figure 2: Evolution of $\ln \|\mathbf{u}^n\|_{\ell^2}$ as a function of $n\Delta t$ for the solution $\mathbf{u}^n=(u_j^n)_{j=0,\cdots,J}$ of the numerical scheme starting from a smooth initial condition with the Dirichlet boundary condition \ref{['schema-Dirichlet']} on the left of the interval and the Neumann condition \ref{['schema-Neumann']} on the right ($k=1$) together with the coefficients given in \ref{['coeff1']}. Here, we have set $J=994$ with $\Delta t =\Delta x=1/(J+1)$.
• Figure 3: Left: space-time evolution of $|u_j^n|$, solution of the numerical scheme with the Dirichlet boundary condition \ref{['schema-Dirichlet']} on the left of the interval and the Neumann condition \ref{['schema-Neumann']} on the right ($k=1$) together with the coefficients given in \ref{['coeff1']}. Right: evolution of $\ln \|\mathbf{u}^n\|_{\ell^2}$ as a function of $n\Delta t$. Here, we have set $J=994$ with $\Delta t=\Delta x=1/(J+1)$.
• Figure 4: Left: amplification factor (dark red curve) $\theta \mapsto \sum_{\ell=-r}^p \, a_\ell \, {\rm e}^{\mathbf{i} \, \ell \, \theta}$ with $r=p=7$ and coefficients $a_\ell$ given in \ref{['coeff2']} within the unit circle $\mathbb{S}^1$ (blue curve). Right: zoom of the amplification factor near the tangency point at $z=1$.
• Figure 5: Left: space-time evolution of $|u_j^n|$, solution of the numerical scheme with the Dirichlet boundary condition \ref{['schema-Dirichlet']} on the left of the interval and the Neumann condition \ref{['schema-Neumann']} on the right ($k=2$) together with the coefficients given in \ref{['coeff2']}. Right: evolution of $\ln \|\mathbf{u}^n\|_{\ell^2}$ as a function of $n\Delta t$. Here, we have set $J=1000$ with $\Delta t=\Delta x=1/(J+1)$.

Theorems & Definitions (2)

• Lemma 1
• proof : Proof of Lemma \ref{['lem1']}