A new multi-step BDF energy stable technique for the extended Fisher-Kolmogorov equation
Abstract
The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher–Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results.
Keyword : extended Fisher-Kolmogorov equation, multi-step BDF method, discrete orthogonal convolution kernels, stability and convergence
How to Cite
Sun, Q., Hu, X., Li, X., Li, Y., & Zhang, L. (2024). A new multi-step BDF energy stable technique for the extended Fisher-Kolmogorov equation. Mathematical Modelling and Analysis, 29(1), 125–140. https://doi.org/10.3846/mma.2024.17430
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
G. Akrivis. Stability of implicit-explicit backward difference formulas for nonlinear parabolic equations. SIAM Journal on Numerical Analysis, 53(1):464–484, 2015. https://doi.org/10.1137/140962619
G. Akrivis and E. Katsoprinakis. Backward difference formulae: new multipliers and stability properties for parabolic equations. Mathematics of Computation, 85(301):2195–2216, 2016. https://doi.org/10.1090/mcom3055
J. Belmonte-Beitia, G.F. Calvo and V.M. Perez-Garcia. Effective particle methods for Fisher–Kolmogorov equations: theory and applications to brain tumor dynamics. Communications in Nonlinear Science and Numerical Simulation, 19(9):3267–3283, 2014. https://doi.org/10.1016/j.cnsns.2014.02.004.
P. Coullet, C. Elphick and D. Repaux. Nature of spatial chaos. Physical review letters, 58(5):431, 1987. https://doi.org/10.1103/PhysRevLett.58.431
P. Danumjaya and A.K. Pani. Numerical methods for the extended FisherKolmogorov (EFK) equation. International Journal of Numerical Analysis and Modeling, 3(2):186–210, 2006.
G.T. Dee and W. van Saarloos. Bistable systems with propagating fronts leading to pattern formation. Physical Review Letters, 60(25):2641, 1988. https://doi.org/10.1103/PhysRevLett.60.2641
U. Grenander, G. Szegö and M. Kac. Toeplitz forms and their applications. Physics Today, 11(10):38–38, 1958. https://doi.org/10.1063/1.3062237
T. Gudi and H.S. Gupta. A fully discrete c0 interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation. Journal of Computational and Applied Mathematics, 247:1–16, 2013. https://doi.org/10.1016/j.cam.2012.12.019
Z. Guozhen. Experiments on director waves in nematic liquid crystals. Physical Review Letters, 49(18):1332, 1982. https://doi.org/10.1103/PhysRevLett.49.1332
R.M. Hornreich, M. Luban and S. Shtrikman. Critical behaviour at the onset of k-space instability at the λ line. Physical Review Letters, 35(18):1678–1681, 1975. https://doi.org/10.1103/PhysRevLett.35.1678
M. Ilati and M. Dehghan. Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation. Engineering with Computers, 34:203–213, 2018. https://doi.org/10.1007/s00366-017-0530-1
K. Ismail, N. Atouani and K. Omrani. A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation. Engineering with Computers, 38:1215–1225, 2021. https://doi.org/10.1007/s00366-020-01269-4
K. Ismail, M. Rahmeni and K. Omrani. An efficient computational approach for solving two-dimensional extended Fisher–Kolmogorov equation. Applicable Analysis, 102(17):4699–4716, 2022. https://doi.org/10.1080/00036811.2022.2134123
T. Kadri and K. Omrani. A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation. Computers & Mathematics with Applications, 61(2):451–459, 2011. https://doi.org/10.1016/j.camwa.2010.11.022
N. Khiari and K. Omrani. Finite difference discretization of the extended Fisher– Kolmogorov equation in two dimensions. Computers & Mathematics with Applications, 62(11):4151–4160, 2011. https://doi.org/10.1016/j.camwa.2011.09.065
X. Li and L. Zhang. Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher–Kolmogorov equation. Applied Numerical Mathematics, 131:39–53, 2018. https://doi.org/10.1016/j.apnum.2018.04.010
H.-L. Liao, B. Ji and L. Zhang. An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA Journal of Numerical Analysis, 42(1):649– 679, 2022. https://doi.org/10.1093/imanum/draa075
H.-L. Liao, Y. Kang and W. Han. Discrete gradient structures of bdf methods up to fifth-order for the phase field crystal model. arXiv preprint arXiv:2201.00609, 2022.
H.-L. Liao, X. Song, T. Tang and T. Zhou. Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection. Science China Mathematics, 64:887–902, 2021. https://doi.org/10.1007/s11425-020-1817-4
H.-L. Liao, T. Tang and T. Zhou. On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen–Cahn equation. SIAM Journal on Numerical Analysis, 58(4):2294–2314, 2020. https://doi.org/10.1137/19M1289157
H.-L. Liao, T. Tang and T. Zhou. Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators. arXiv preprint arXiv:2011.13383, 2020.
H.-L. Liao, T. Tang and T. Zhou. A new discrete energy technique for multi-step backward difference formulas. arXiv preprint arXiv:2102.04644, 2021.
H.-L. Liao and Z. Zhang. Analysis of adaptive BDF2 scheme for diffusion equations. Mathematics of Computation, 90(329):1207–1226, 2021. https://doi.org/10.1090/mcom/3585
J. Liu. Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains. SIAM Journal on Numerical Analysis, 51(2):743–772, 2013. https://doi.org/10.1137/110825996
W. Van Saarloos. Front propagation into unstable states. II. linear versus nonlinear marginal stability and rate of convergence. Physical Review A, 39(12):6367, 1989. https://doi.org/10.1103/PhysRevA.39.6367
Q. Sun, B. Ji and L. Zhang. A convex splitting BDF2 method with variable time-steps for the extended Fisher–Kolmogorov equation. Computers & Mathematics with Applications, 114:73–82, 2022. https://doi.org/10.1016/j.camwa.2022.03.017
V. Thomée. Galerkin finite element methods for parabolic problems. Lecture notes in mathematics, 1054, 1984.
J. Xu, Y. Li, S. Wu and A. Bousquet. On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Computer Methods in Applied Mechanics and Engineering, 345:826–853, 2019. https://doi.org/10.1016/j.cma.2018.09.017
Y.L. Zhou. Application of discrete functional analysis to the finite difference method. Inter. Acad. Publishers, Beijing, 1990.
G. Akrivis and E. Katsoprinakis. Backward difference formulae: new multipliers and stability properties for parabolic equations. Mathematics of Computation, 85(301):2195–2216, 2016. https://doi.org/10.1090/mcom3055
J. Belmonte-Beitia, G.F. Calvo and V.M. Perez-Garcia. Effective particle methods for Fisher–Kolmogorov equations: theory and applications to brain tumor dynamics. Communications in Nonlinear Science and Numerical Simulation, 19(9):3267–3283, 2014. https://doi.org/10.1016/j.cnsns.2014.02.004.
P. Coullet, C. Elphick and D. Repaux. Nature of spatial chaos. Physical review letters, 58(5):431, 1987. https://doi.org/10.1103/PhysRevLett.58.431
P. Danumjaya and A.K. Pani. Numerical methods for the extended FisherKolmogorov (EFK) equation. International Journal of Numerical Analysis and Modeling, 3(2):186–210, 2006.
G.T. Dee and W. van Saarloos. Bistable systems with propagating fronts leading to pattern formation. Physical Review Letters, 60(25):2641, 1988. https://doi.org/10.1103/PhysRevLett.60.2641
U. Grenander, G. Szegö and M. Kac. Toeplitz forms and their applications. Physics Today, 11(10):38–38, 1958. https://doi.org/10.1063/1.3062237
T. Gudi and H.S. Gupta. A fully discrete c0 interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation. Journal of Computational and Applied Mathematics, 247:1–16, 2013. https://doi.org/10.1016/j.cam.2012.12.019
Z. Guozhen. Experiments on director waves in nematic liquid crystals. Physical Review Letters, 49(18):1332, 1982. https://doi.org/10.1103/PhysRevLett.49.1332
R.M. Hornreich, M. Luban and S. Shtrikman. Critical behaviour at the onset of k-space instability at the λ line. Physical Review Letters, 35(18):1678–1681, 1975. https://doi.org/10.1103/PhysRevLett.35.1678
M. Ilati and M. Dehghan. Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation. Engineering with Computers, 34:203–213, 2018. https://doi.org/10.1007/s00366-017-0530-1
K. Ismail, N. Atouani and K. Omrani. A three-level linearized high-order accuracy difference scheme for the extended Fisher–Kolmogorov equation. Engineering with Computers, 38:1215–1225, 2021. https://doi.org/10.1007/s00366-020-01269-4
K. Ismail, M. Rahmeni and K. Omrani. An efficient computational approach for solving two-dimensional extended Fisher–Kolmogorov equation. Applicable Analysis, 102(17):4699–4716, 2022. https://doi.org/10.1080/00036811.2022.2134123
T. Kadri and K. Omrani. A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation. Computers & Mathematics with Applications, 61(2):451–459, 2011. https://doi.org/10.1016/j.camwa.2010.11.022
N. Khiari and K. Omrani. Finite difference discretization of the extended Fisher– Kolmogorov equation in two dimensions. Computers & Mathematics with Applications, 62(11):4151–4160, 2011. https://doi.org/10.1016/j.camwa.2011.09.065
X. Li and L. Zhang. Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher–Kolmogorov equation. Applied Numerical Mathematics, 131:39–53, 2018. https://doi.org/10.1016/j.apnum.2018.04.010
H.-L. Liao, B. Ji and L. Zhang. An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA Journal of Numerical Analysis, 42(1):649– 679, 2022. https://doi.org/10.1093/imanum/draa075
H.-L. Liao, Y. Kang and W. Han. Discrete gradient structures of bdf methods up to fifth-order for the phase field crystal model. arXiv preprint arXiv:2201.00609, 2022.
H.-L. Liao, X. Song, T. Tang and T. Zhou. Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection. Science China Mathematics, 64:887–902, 2021. https://doi.org/10.1007/s11425-020-1817-4
H.-L. Liao, T. Tang and T. Zhou. On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen–Cahn equation. SIAM Journal on Numerical Analysis, 58(4):2294–2314, 2020. https://doi.org/10.1137/19M1289157
H.-L. Liao, T. Tang and T. Zhou. Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators. arXiv preprint arXiv:2011.13383, 2020.
H.-L. Liao, T. Tang and T. Zhou. A new discrete energy technique for multi-step backward difference formulas. arXiv preprint arXiv:2102.04644, 2021.
H.-L. Liao and Z. Zhang. Analysis of adaptive BDF2 scheme for diffusion equations. Mathematics of Computation, 90(329):1207–1226, 2021. https://doi.org/10.1090/mcom/3585
J. Liu. Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains. SIAM Journal on Numerical Analysis, 51(2):743–772, 2013. https://doi.org/10.1137/110825996
W. Van Saarloos. Front propagation into unstable states. II. linear versus nonlinear marginal stability and rate of convergence. Physical Review A, 39(12):6367, 1989. https://doi.org/10.1103/PhysRevA.39.6367
Q. Sun, B. Ji and L. Zhang. A convex splitting BDF2 method with variable time-steps for the extended Fisher–Kolmogorov equation. Computers & Mathematics with Applications, 114:73–82, 2022. https://doi.org/10.1016/j.camwa.2022.03.017
V. Thomée. Galerkin finite element methods for parabolic problems. Lecture notes in mathematics, 1054, 1984.
J. Xu, Y. Li, S. Wu and A. Bousquet. On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Computer Methods in Applied Mechanics and Engineering, 345:826–853, 2019. https://doi.org/10.1016/j.cma.2018.09.017
Y.L. Zhou. Application of discrete functional analysis to the finite difference method. Inter. Acad. Publishers, Beijing, 1990.