# American Institute of Mathematical Sciences

February  2020, 13(1): 1-32. doi: 10.3934/krm.2020001

## The Neumann numerical boundary condition for transport equations

 1 Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France 2 Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan (CNRS UMR5208), 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

* Corresponding author: Frédéric Lagoutière

Received  October 2018 Revised  June 2019 Published  December 2019

Fund Project: Both authors are supported by the ANR project BoND, ANR-13-BS01-0009, and by the ANR project NABUCO, ANR-17-CE40-0025

In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in $\ell^\infty$ 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.

Citation: Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic & Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001
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##### References:
The mesh on $\mathbb{R}^+ \times (0,L)$ in blue, and the "ghost cells" in red ($r = p = 2$ here)
Top: updating iteratively the ghost values at the outflow boundary ($r = p = k_b = 2$). Bottom: updating the numerical approximation in the interior
Initial condition $u^{0,2}$ with $40$ cells in $(0, 1)$
Numerical and exact solutions at time $t = 0.2625$ with initial condition $u^{0,2}$ ($40$ cells)
Numerical and exact solutions at time $t = 0.5075$ with initial condition $u^{0,2}$ ($40$ cells)
Error for datum $u^{0,1}$. The result presented in this paper guarantees that the order is at least $3/2$ when $k_b = 2$, and at least $1/2$ when $k_b = 1$. The "order" that is indicated in bracket is a "local" order computed, for each line $k$, as $-\log(e_k/e_{k-1})/\log(2)$
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.53 \times 10^{-3}$ $8.34 \times 10^{-3}$ 20 $8.28 \times 10^{-4}$ (order 1.61) $4.92 \times 10^{-3}$ (order 0.76) 40 $2.31 \times 10^{-4}$ (order 1.84) $2.63 \times 10^{-3}$ (order 0.90) 80 $6.09 \times 10^{-5}$ (order 1.93) $1.40 \times 10^{-3}$ (order 0.91) 160 $1.56 \times 10{-5}$ order 1.96) $7.21 \times 10^{-4}$ (order 0.96) 320 $3.97 \times 10^{-6}$ (order 1.97) $3.66 \times 10^{-4}$ (order 0.98) 640 $1.00 \times 10^{-6}$ (order 1.99) $1.84 \times 10^{-4}$ (order 0.99) 1280 $2.52 \times 10^{-7}$ (order 1.99) $9.24 \times 10^{-5}$ (order 0.99)
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.53 \times 10^{-3}$ $8.34 \times 10^{-3}$ 20 $8.28 \times 10^{-4}$ (order 1.61) $4.92 \times 10^{-3}$ (order 0.76) 40 $2.31 \times 10^{-4}$ (order 1.84) $2.63 \times 10^{-3}$ (order 0.90) 80 $6.09 \times 10^{-5}$ (order 1.93) $1.40 \times 10^{-3}$ (order 0.91) 160 $1.56 \times 10{-5}$ order 1.96) $7.21 \times 10^{-4}$ (order 0.96) 320 $3.97 \times 10^{-6}$ (order 1.97) $3.66 \times 10^{-4}$ (order 0.98) 640 $1.00 \times 10^{-6}$ (order 1.99) $1.84 \times 10^{-4}$ (order 0.99) 1280 $2.52 \times 10^{-7}$ (order 1.99) $9.24 \times 10^{-5}$ (order 0.99)
Error for datum $u^{0,2}$. The result presented in this paper guarantees that the order is at least $3/2$ when $k_b = 2$, and at least $1/2$ when $k_b = 1$. Concerning the order, see Table 1
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.80 \times 10^{-3}$ $1.03 \times 10^{-2}$ 20 $8.25 \times 10^{-4}$ (order 1.76) $5.78 \times 10^{-3}$ (order 0.83) 40 $2.53 \times 10^{-4}$ (order 1.71) $3.09 \times 10^{-3}$ (order 0.91) 80 $7.81 \times 10^{-5}$ (order 1.69) $1.62 \times 10^{-3}$ (order 0.93) 160 $2.36 \times 10^{-5}$ (order 1.73) $8.29 \times 10^{-4}$ (order 0.97) 320 $7.11 \times 10^{-6}$ (order 1.73) $4.19 \times 10^{-4}$ (order 0.98) 640 $2.14 \times 10^{-6}$ (order 1.74) $2.11 \times 10^{-4}$ (order 0.99) 1280 $6.43 \times 10^{-7}$ (order 1.73) $1.06 \times 10^{-4}$ (order 0.99)
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.80 \times 10^{-3}$ $1.03 \times 10^{-2}$ 20 $8.25 \times 10^{-4}$ (order 1.76) $5.78 \times 10^{-3}$ (order 0.83) 40 $2.53 \times 10^{-4}$ (order 1.71) $3.09 \times 10^{-3}$ (order 0.91) 80 $7.81 \times 10^{-5}$ (order 1.69) $1.62 \times 10^{-3}$ (order 0.93) 160 $2.36 \times 10^{-5}$ (order 1.73) $8.29 \times 10^{-4}$ (order 0.97) 320 $7.11 \times 10^{-6}$ (order 1.73) $4.19 \times 10^{-4}$ (order 0.98) 640 $2.14 \times 10^{-6}$ (order 1.74) $2.11 \times 10^{-4}$ (order 0.99) 1280 $6.43 \times 10^{-7}$ (order 1.73) $1.06 \times 10^{-4}$ (order 0.99)
Error for datum $u^{0,3}$. The result presented in this paper does not apply to the case $k_b = 2$, and guarantees that the convergence order is at least at least $1/2$ when $k_b = 1$. Concerning the order, see Table 1
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.84 \times 10^{-3}$ $1.08 \times 10^{-2}$ 20 $9.18 \times 10^{-4}$ (order 1.63) $6.00 \times 10^{-3}$ (order 0.85) 40 $3.02 \times 10^{-4}$ (order 1.60) $3.20 \times 10^{-3}$ (order 0.91) 80 $9.76 \times 10^{-5}$ (order 1.63) $1.67 \times 10^{-3}$ (order 0.93) 160 $3.08 \times 10^{-5}$ (order 1.66) $8.56 \times 10^{-4}$ (order 0.97) 320 $9.72 \times 10^{-6}$ (order 1.66) $4.32 \times 10^{-4}$ (order 0.98) 640 $3.08 \times 10^{-6}$ (order 1.66) $2.17 \times 10^{-4}$ (order 0.99) 1280 $9.71 \times 10^{-7}$ (order 1.67) $1.09 \times 10^{-4}$ (order 0.99)
 Number of cells $J$ Measured error with $k_b = 2$ Measured error with $k_b = 1$ 10 $2.84 \times 10^{-3}$ $1.08 \times 10^{-2}$ 20 $9.18 \times 10^{-4}$ (order 1.63) $6.00 \times 10^{-3}$ (order 0.85) 40 $3.02 \times 10^{-4}$ (order 1.60) $3.20 \times 10^{-3}$ (order 0.91) 80 $9.76 \times 10^{-5}$ (order 1.63) $1.67 \times 10^{-3}$ (order 0.93) 160 $3.08 \times 10^{-5}$ (order 1.66) $8.56 \times 10^{-4}$ (order 0.97) 320 $9.72 \times 10^{-6}$ (order 1.66) $4.32 \times 10^{-4}$ (order 0.98) 640 $3.08 \times 10^{-6}$ (order 1.66) $2.17 \times 10^{-4}$ (order 0.99) 1280 $9.71 \times 10^{-7}$ (order 1.67) $1.09 \times 10^{-4}$ (order 0.99)
Spectral radii and $l^2$ induced norms of the linear operator associated with the scheme
 $J$ Spectral radius, $k_b = 1$ $l^2$ norm, $k_b = 1$ Spectral radius, $k_b = 2$ $l^2$ norm, $k_b = 2$ 20 0.7100 0.9999 0.7098 1.0035 80 0.74300 0.9999 0.7513 1.0035 320 0.9208 0.9999 0.9212 1.0035 1280 0.9817 0.9999 0.9805 1.0035
 $J$ Spectral radius, $k_b = 1$ $l^2$ norm, $k_b = 1$ Spectral radius, $k_b = 2$ $l^2$ norm, $k_b = 2$ 20 0.7100 0.9999 0.7098 1.0035 80 0.74300 0.9999 0.7513 1.0035 320 0.9208 0.9999 0.9212 1.0035 1280 0.9817 0.9999 0.9805 1.0035
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