# 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 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.
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
 [1] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [2] Zhenghuan Gao, Peihe Wang. Global $C^2$-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1201-1223. doi: 10.3934/dcds.2021152 [3] Guillaume Bal, Alexandre Jollivet. Boundary control for transport equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022014 [4] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [5] Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230 [6] Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190 [7] Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297 [8] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [9] Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153 [10] Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 [11] G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 [12] Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure and Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761 [13] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [14] Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935 [15] Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 [16] Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004 [17] Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 [18] Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129 [19] Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048 [20] Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic and Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865

2021 Impact Factor: 1.398