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

The Neumann numerical boundary condition for transport equations

Jean-François Coulombel 1, and  Frédéric Lagoutière 2,,

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.

Keywords: Transport equations, numerical schemes, Neumann boundary condition, stability, convergence.
Mathematics Subject Classification: Primary: 65M12, 65M06, 65M20.
Citation: Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic and Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001
References:
[1]

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729-796. 

[2]

B. Boutin and J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems: Numerical boundary layers, Numer. Math. Theory Methods Appl., 10 (2017), 489-519.  doi: 10.4208/nmtma.2017.m1525.

[3]

C. Chainais-Hillairet and E. Grenier, Numerical boundary layers for hyperbolic systems in 1-D, M2AN Math. Model. Numer. Anal., 35 (2001), 91-106.  doi: 10.1051/m2an:2001100.

[4]

J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, in HCDTE Lecture Notes. Part Ⅰ. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, American Institute of Mathematical Sciences, 6 (2013), 146pp.

[5]

J.-F. Coulombel, Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis, Ann. Fac. Sci. Toulouse Math. (6), 28 (2019), 259-327.  doi: 10.5802/afst.1600.

[6]

J.-F. Coulombel and A. Gloria, Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., 80 (2011), 165-203.  doi: 10.1090/S0025-5718-10-02368-9.

[7]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.

[8]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques. Tome 3, Collection du Commissariat à l'Énergie Atomique: Série Scientifique., Masson, Paris, 1985.

[9]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.1090/S0025-5718-1977-0436612-4.

[10]

M. Goldberg, On a boundary extrapolation theorem by Kreiss, Math. Comp., 31 (1977), 469-477.  doi: 10.1090/S0025-5718-1977-0443363-9.

[11]

M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. Ⅰ, Math. Comp., 32 (1978), 1097-1107.  doi: 10.1090/S0025-5718-1978-0501998-X.

[12]

M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. Ⅱ, Math. Comp., 36 (1981), 603-626.  doi: 10.1090/S0025-5718-1981-0606519-9.

[13]

B. Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29 (1975), 396-406.  doi: 10.1090/S0025-5718-1975-0386296-7.

[14]

B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley & Sons, 1995.

[15]

B. Gustafsson, H.-O. Kreiss and A. Sundström, Stability theory of difference approximations for mixed initial boundary value problems. Ⅱ, Math. Comp., 26 (1972), 649-686.  doi: 10.1090/S0025-5718-1972-0341888-3.

[16]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, in Acta Numerica, 1999, vol. 8 of Acta Numer., Cambridge Univ. Press, 1999, 47–106. doi: 10.1017/S0962492900002890.

[17]

L. Halpern, Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation, Math. Comp., 38 (1982), 415-429.  doi: 10.1090/S0025-5718-1982-0645659-6.

[18]

G. W. Hedstrom, Norms of powers of absolutely convergent Fourier series, Michigan Math. J., 13 (1966), 393-416.  doi: 10.1307/mmj/1028999598.

[19]

R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47 (1986), 437-459.  doi: 10.2307/2008166.

[20]

R. L. Higdon, Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal., 31 (1994), 64-100.  doi: 10.1137/0731004.

[21]

H.-O. Kreiss, Difference approximations for hyperbolic differential equations, in Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, 1966, 51–58.

[22]

H.-O. Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. Ⅰ, Math. Comp., 22 (1968), 703-714.  doi: 10.1090/S0025-5718-1968-0241010-7.

[23]

H.-O. Kreiss and E. Lundqvist, On difference approximations with wrong boundary values, Math. Comp., 22 (1968), 1-12.  doi: 10.1090/S0025-5718-1968-0228193-X.

[24]

S. Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc., 137 (1969), 177-201.  doi: 10.1090/S0002-9947-1969-0237982-4.

[25]

G. Strang, Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., 41 (1962), 147-154.  doi: 10.1002/sapm1962411147.

[26]

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), 2004. doi: 10.1137/1.9780898717938.

[27]

V. Thomée, Stability of difference schemes in the maximum-norm, J. Differential Equations, 1 (1965), 273-292.  doi: 10.1016/0022-0396(65)90008-2.

[28]

L. N. Trefethen, Instability of difference models for hyperbolic initial-boundary value problems, Comm. Pure Appl. Math., 37 (1984), 329-367.  doi: 10.1002/cpa.3160370305.

[29]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, 2005, The behavior of nonnormal matrices and operators.

[30]

L. Wu, The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., 64 (1995), 71-88.  doi: 10.2307/2153323.

show all references

References:
[1]

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729-796. 

[2]

B. Boutin and J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems: Numerical boundary layers, Numer. Math. Theory Methods Appl., 10 (2017), 489-519.  doi: 10.4208/nmtma.2017.m1525.

[3]

C. Chainais-Hillairet and E. Grenier, Numerical boundary layers for hyperbolic systems in 1-D, M2AN Math. Model. Numer. Anal., 35 (2001), 91-106.  doi: 10.1051/m2an:2001100.

[4]

J.-F. Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, in HCDTE Lecture Notes. Part Ⅰ. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, American Institute of Mathematical Sciences, 6 (2013), 146pp.

[5]

J.-F. Coulombel, Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis, Ann. Fac. Sci. Toulouse Math. (6), 28 (2019), 259-327.  doi: 10.5802/afst.1600.

[6]

J.-F. Coulombel and A. Gloria, Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., 80 (2011), 165-203.  doi: 10.1090/S0025-5718-10-02368-9.

[7]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.

[8]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques. Tome 3, Collection du Commissariat à l'Énergie Atomique: Série Scientifique., Masson, Paris, 1985.

[9]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.1090/S0025-5718-1977-0436612-4.

[10]

M. Goldberg, On a boundary extrapolation theorem by Kreiss, Math. Comp., 31 (1977), 469-477.  doi: 10.1090/S0025-5718-1977-0443363-9.

[11]

M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. Ⅰ, Math. Comp., 32 (1978), 1097-1107.  doi: 10.1090/S0025-5718-1978-0501998-X.

[12]

M. Goldberg and E. Tadmor, Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. Ⅱ, Math. Comp., 36 (1981), 603-626.  doi: 10.1090/S0025-5718-1981-0606519-9.

[13]

B. Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29 (1975), 396-406.  doi: 10.1090/S0025-5718-1975-0386296-7.

[14]

B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley & Sons, 1995.

[15]

B. Gustafsson, H.-O. Kreiss and A. Sundström, Stability theory of difference approximations for mixed initial boundary value problems. Ⅱ, Math. Comp., 26 (1972), 649-686.  doi: 10.1090/S0025-5718-1972-0341888-3.

[16]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, in Acta Numerica, 1999, vol. 8 of Acta Numer., Cambridge Univ. Press, 1999, 47–106. doi: 10.1017/S0962492900002890.

[17]

L. Halpern, Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation, Math. Comp., 38 (1982), 415-429.  doi: 10.1090/S0025-5718-1982-0645659-6.

[18]

G. W. Hedstrom, Norms of powers of absolutely convergent Fourier series, Michigan Math. J., 13 (1966), 393-416.  doi: 10.1307/mmj/1028999598.

[19]

R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47 (1986), 437-459.  doi: 10.2307/2008166.

[20]

R. L. Higdon, Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal., 31 (1994), 64-100.  doi: 10.1137/0731004.

[21]

H.-O. Kreiss, Difference approximations for hyperbolic differential equations, in Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, 1966, 51–58.

[22]

H.-O. Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. Ⅰ, Math. Comp., 22 (1968), 703-714.  doi: 10.1090/S0025-5718-1968-0241010-7.

[23]

H.-O. Kreiss and E. Lundqvist, On difference approximations with wrong boundary values, Math. Comp., 22 (1968), 1-12.  doi: 10.1090/S0025-5718-1968-0228193-X.

[24]

S. Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc., 137 (1969), 177-201.  doi: 10.1090/S0002-9947-1969-0237982-4.

[25]

G. Strang, Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., 41 (1962), 147-154.  doi: 10.1002/sapm1962411147.

[26]

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), 2004. doi: 10.1137/1.9780898717938.

[27]

V. Thomée, Stability of difference schemes in the maximum-norm, J. Differential Equations, 1 (1965), 273-292.  doi: 10.1016/0022-0396(65)90008-2.

[28]

L. N. Trefethen, Instability of difference models for hyperbolic initial-boundary value problems, Comm. Pure Appl. Math., 37 (1984), 329-367.  doi: 10.1002/cpa.3160370305.

[29]

L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, 2005, The behavior of nonnormal matrices and operators.

[30]

L. Wu, The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., 64 (1995), 71-88.  doi: 10.2307/2153323.

Figure 1.  The mesh on $ \mathbb{R}^+ \times (0,L) $ in blue, and the "ghost cells" in red ($ r = p = 2 $ here)
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Figure 2.  Top: updating iteratively the ghost values at the outflow boundary ($ r = p = k_b = 2 $). Bottom: updating the numerical approximation in the interior
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Figure 3.  Initial condition $ u^{0,2} $ with $ 40 $ cells in $ (0, 1) $
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Figure 4.  Numerical and exact solutions at time $ t = 0.2625 $ with initial condition $ u^{0,2} $ ($ 40 $ cells)
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Figure 5.  Numerical and exact solutions at time $ t = 0.5075 $ with initial condition $ u^{0,2} $ ($ 40 $ cells)
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Table 1.  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)
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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)
Table 2.  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)
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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)
Table 3.  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)
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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)
Table 4.  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
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$ 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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