×
Gao, Ruimin; Li, Dongfang; Mei, Ming; Zhao, Dan

A decoupled, linearly implicit and high-order structure-preserving scheme for Euler-Poincaré equations. (English) Zbl 1540.65291

Math. Comput. Simul. 218, 679-703 (2024).

Cited in 2 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Keywords:

decoupled scheme; high-order accuracy; modified two-component Euler-Poincaré equations; mass conservation; energy stability
Cite Review PDF
Full Text: DOI

References:

[1] Antonopoulos, D. C.; Dougalis, V. A.; Mitsotakis, D. E., Error estimates for Galerkin finite element methods for the Camassa-Holm equation. Numer. Math., 833-862 (2019) · Zbl 1419.65050
[2] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal., 215-239 (2007) · Zbl 1105.76013
[3] Cao, W.; Li, D.; Zhang, Z., Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations. Sci. China Math., 1731-1748 (2022) · Zbl 1493.65147
[4] Cao, H.; Sun, Z.; Gao, G., A three-level linearized finite difference scheme for the Camassa-Holm equation. Numer. Methods Partial Differential Equations, 451-471 (2014) · Zbl 1290.65071
[5] Chertock, A.; Kurganov, A.; Liu, Y., Finite-volume-particle methods for the two-component Camassa-Holm system. Commun. Comput. Phys., 480-502 (2019) · Zbl 1473.65135
[6] Coclite, G.; Karlsen, K.; Risebro, N., A convergent finite difference scheme for the Camassa-Holm equation with general \(H^1\) initial data. SIAM J. Numer. Anal., 1554-1579 (2008) · Zbl 1172.35310
[7] Cohen, D.; Matsuo, T.; Raynaud, X., A multi-symplectic numerical integrator for the two-component Camassa-Holm equation. J. Nonlinear Math. Phys., 442-453 (2021) · Zbl 1420.35289
[8] Cohen, D.; Owren, B.; Raynaud, X., Multi-symplectic integration of the Camassa-Holm equation. J. Comput. Phys., 5492-5512 (2008) · Zbl 1148.65093
[9] Escher, J.; Kohlmann, M.; Lenells, J., The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations. J. Geom. Phys., 436-452 (2011) · Zbl 1210.58007
[10] Escher, J.; Lechtenfeld, O.; Yin, Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete Contin. Dyn. Syst., 493-513 (2007) · Zbl 1149.35307
[11] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica D, 229-243 (1996) · Zbl 0900.35345
[12] Ghiloufi, A.; Omrani, K., New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer. Methods Partial Differential Equations, 451-500 (2018) · Zbl 1390.65068
[13] Gong, Y.; Hong, Q.; Wang, C.; Wang, Y., Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation based on the quadratic auxiliary variable approach. Adv. Appl. Math. Mech., 1233-1255 (2023) · Zbl 1524.65650
[14] Gong, Y.; Wang, Y., An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs. Commun. Comput. Phys., 1313-1339 (2016) · Zbl 1388.65120
[15] Gu, X.; Cai, W.; Wang, Y.; Jiang, C., Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator. IMA J. Numer. Anal. (2023)
[16] Gunzburger, M.; Li, B.; Wang, J.; Yang, Z., A mass conservative, well balanced, tangency preserving and energy decaying method for the shallow water equations on a sphere. J. Comput. Phys., 1-18 (2022) · Zbl 1515.76096
[17] Holdahl, R.; Holden, H.; Lie, K., Unconditionally stable splitting methods for the shallow water equations. BIT Numer. Math., 451-472 (1999) · Zbl 0945.76059
[18] Holden, H.; Raynaud, X., Convergence of a finite difference scheme for the Camassa-Holm equation. SIAM J. Numer. Anal., 1655-1680 (2006) · Zbl 1122.76065
[19] Holden, H.; Raynaud, X., Periodic conservative solutions of the Camassa-Holm equation. Ann. Inst. Fourier, 945-988 (2008) · Zbl 1158.35079
[20] Holm, D.; Naraigh, L.; Tronci, C., Singular solutions of a modified two-component Camassa-Holm equation. Phys. Rev. E, 1-13 (2009)
[21] Hong, Q.; Gong, Y.; Lv, Z., Linear and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa-Holm equation. Appl. Math. Comput., 86-95 (2019) · Zbl 1429.65241
[22] Hong, Q.; Gong, Y.; Zhao, J., A physics-informed structure-preserving numerical scheme for the phase-field hydrodynamic model of ternary fluid flows. Numer. Math.: Theory Methods Appl., 3, 565-596 (2023) · Zbl 07814745
[23] Jiang, C.; Gong, Y.; Cai, W.; Wang, Y., A linearly implicit structure-preserving scheme for the Camassa-Holm equation based on multiple scalar auxiliary variables approach. J. Sci. Comput., 20, 1-20 (2020) · Zbl 1436.65104
[24] Jiang, C.; Wang, Y.; Gong, Y., Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. Appl. Numer. Math., 85-97 (2020) · Zbl 1439.37079
[25] Johnson, R. S., Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech., 63-82 (2002) · Zbl 1037.76006
[26] Lakshmanan, M., Integrable nonlinear wave equations and possible connections to tsunami dynamics. Tsunami Nonlinear Waves, 31-49 (2007) · Zbl 1310.76044
[27] Li, D.; Li, X., Relaxation exponential rosenbrock-type methods for oscillatory Hamiltonian systems. SIAM J. Sci. Comput., 6, A2886-A2911 (2000) · Zbl 1537.65194
[28] Li, D.; Li, X.; Sun, H., Optimal error estimates of SAV Crank-Nicolson finite element method for the coupled nonlinear Schrödinger equation. J. Sci. Comput., 3, 71 (2023) · Zbl 07766141
[29] Li, D.; Li, X.; Zhang, Z., Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs. Math. Comp., 339, 117-146 (2023) · Zbl 1511.65068
[30] Li, D.; Li, X.; Zhang, Z., Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systems. J. Comput. Phys., 1-19 (2023) · Zbl 07652817
[31] Li, X.; Wen, J.; Li, D., Mass- and energy-conserving difference schemes for nonlinear fractional Schrödinger equations. Appl. Math. Lett., 1-7 (2021) · Zbl 1450.65081
[32] Liu, H.; Xing, Y., An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Sci. Comput., 1919-1934 (2016) · Zbl 1382.65321
[33] Matsuo, T., A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation. J. Comput. Appl. Math., 1258-1266 (2010) · Zbl 1203.65185
[34] Matsuo, T.; Yamaguchi, H., An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations. J. Comput. Phys., 4346-4358 (2009) · Zbl 1169.65097
[35] McLachlan, R.; Zhang, X., Asymptotic blow-up profiles for modified Camassa-Holm equations. SIAM J. Appl. Dyn. Syst., 452-468 (2011) · Zbl 1234.35208
[36] Novikov, V., Generalizations of the Camassa-Holm equation. J. Phys. A, 232-238 (2009)
[37] Olver, P.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E, 1900-1906 (1996)
[38] Ranocha, H.; Mitsotakis, D.; Ketcheson, D., A broad class of conservative numerical methods for dispersive wave equations. Commun. Comput. Phys., 979-1029 (2021) · Zbl 1473.65154
[39] Ranocha, H.; Sayyari, M.; Dalcin, L.; Parsani, M.; Ketcheson, D., Relaxation Runge-Kutta methods: fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM J. Sci. Comput., 612-638 (2020) · Zbl 1432.76207
[40] Raphaël, D., A note on well-posedness for Camassa-Holm equation. J. Differential Equations, 429-444 (2003) · Zbl 1048.35076
[41] Tao, N., A decoupled and conservative difference scheme with fourth-order accuracy for the symmetric regularized long wave equations. Appl. Math. Comput., 9461-9648 (2013) · Zbl 1290.76104
[42] Wang, D.; Xiao, A.; Yang, W., Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys., 670-681 (2013) · Zbl 1297.65100
[43] Xu, Y.; Zhao, P.; Ye, Z.; Zheng, Z., Conservative second-order finite difference method for Camassa-Holm equation with periodic boundary condition. Int. J. Comput. Math. (2023)
[44] Yan, K.; Liu, Y., The initial-value problem to the modified two-component Euler-Poincaré equations. SIAM J. Math. Anal., 2006-2039 (2022) · Zbl 1492.35211
[45] Yang, X., Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys., 294-316 (2016) · Zbl 1373.82106
[46] Yang, X.; Zhao, J.; He, X., Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J. Comput. Appl. Math., 80-97 (2018) · Zbl 1462.65117
[47] Yu, C.; Feng, B.; Sheu, T., Numerical solutions to a two-component Camassa-Holm equation. J. Comput. Appl. Math., 317-337 (2018) · Zbl 1429.76076
[48] Zhang, Q.; Liu, L., Convergence and stability in maximum norms of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers’ equation. J. Sci. Comput., 59 (2021) · Zbl 1468.35178
[49] Zhang, Q.; Liu, L.; Zhang, Z., Linearly implicit invariant-preserving decoupled difference scheme for the rotation-two-component Camassa-Holm system. SIAM J. Sci. Comput., 2226-2252 (2022) · Zbl 1492.65248
[50] Zhang, Q.; Qin, Y.; Sun, Z., Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity. Numer. Algorithms, 1081-1114 (2022) · Zbl 1500.65049
[51] Zhang, Q.; Yan, T.; Gao, G., The energy method for high-order invariants in shallow water wave equations. Appl. Math. Lett. (2023) · Zbl 1530.35237
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.