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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 |
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.
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. |




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Measured error with |
Measured error with |
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Number of cells |
Measured error with |
Measured error with |
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Number of cells |
Measured error with |
Measured error with |
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640 | ||
1280 |
Spectral radius, |
Spectral radius, |
|||
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 |
Spectral radius, |
Spectral radius, |
|||
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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