Abstract
In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.
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Akrivis G, Li B, Li D. Energy-decaying extrapolated RK-SAV methods for the Allen-Cahn and Cahn-Hilliard equations. SIAM J Sci Comput, 2019, 41: A3703–A3727
Bellman R. The stability of solutions of linear differential equations. Duke Math J, 1943, 10: 643–647
Brugnano L, Caccia G F, Iavernaro F. Energy conservation issues in the numerical solution of the semilinear wave equation. Appl Math Comput, 2015, 270: 842–870
Cai J, Shen J. Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs. J Comput Phys, 2020, 401: 108975
Cai W, Jiang C, Wang Y, et al. Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions. J Comput Phys, 2019, 395: 166–185
Cannon J R, Lin Y. Non-classical H1 projection and Galerkin methods for non-linear parabolic integro-differential equations. Calcolo, 1988, 25: 187–201
Cano B, Moreta M J. Multistep cosine methods for second-order partial differential systems. IMA J Numer Anal, 2010, 30: 431–461
Cao W, Li D, Zhang Z. Optimal superconvergence of energy conserving local discontinuous Galerkin methods for wave equations. Commun Comput Phys, 2017, 21: 211–236
Carstensen C, Dolzmann G. Time-Space discretization of the nonlinear hyperbolic system utt = div(σ(DU) + DUt). SIAM J Numer Anal, 2004, 42: 75–89
Dendy J E. Galerkin’s method for some highly nonlinear problems. SIAM J Numer Anal, 1977, 14: 327–347
Dodd R K, Eilbeck I C, Gibbon J D, et al. Solitons and Nonlinear Wave Equations. London-New York: Academic Press, 1982
Drazin P J, Johnson R S. Solitons: An introduction. Physics Today, 1990, 43: 70–71
Farago I. Finite element method for solving nonlinear parabolic equations. Comput Math Appl, 1991, 21: 59–69
Garcia S. Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: The discrete-time case. Numer Methods Partial Differential Equations, 1994, 10: 149–169
Gauckler L. Error analysis of trigonometric integrators for semilinear wave equations. SIAM J Numer Anal, 2015, 53: 1082–1106
Grimm V. A note on the Gautschi-type method for oscillatory second-order differential equations. Numer Math, 2005, 102: 61–66
Heywood J G, Rannacher R. Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization. SIAM J Numer Anal, 1990, 27: 353–384
Li B, Gao H, Sun W. Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J Numer Anal, 2014, 52: 933–954
Li B, Sun W. Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int J Numer Anal Model, 2013, 10: 622–633
Li B, Sun W. Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J Numer Anal, 2013, 51: 1959–1977
Li D, Sun W. Linearly implicit and high-order energy-conserving schemes for nonlinear wave equations. J Sci Comput, 2020, 83: 65
Li D, Wang J, Zhang J. Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J Sci Comput, 2017, 39: A3067–A3088
Li X, Wen J, Li D, Mass- and energy-conserving difference schemes for nonlinear fractional Schrödinger equations. Appl Math Lett, 2021, 111: 106686
Li D, Wu C, Zhang Z. Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J Sci Comput, 2019, 80: 403–419
Luskin M. A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J Numer Anal, 1979, 16: 284–299
Rachford Jr H H. Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations. SIAM J Numer Anal, 1973, 10: 1010–1026
Roger T. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. New York: Springer, 1997
Sassaman R, Biswas A. Soliton perturbation theory for phi-four model and nonlinear Klein-Gordon equations. Commun Nonlinear Sci Numer Simul, 2009, 14: 3239–3249
Shen J, Xu J. Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J Numer Anal, 2018, 56: 2895–2912
Shen J, Xu J, Yang J. The scalar auxiliary variable (SAV) approach for gradient flows. J Comput Phys, 2018, 353: 407–416
Shen J, Xu J, Yang J. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev, 2019, 61: 474–506
Thomée V. Galerkin Finite Element Methods for Parabolic Problems. Berlin-Heidelberg: Springer-Verlag, 1997
Wang B, Wu X. The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein-Gordon equations. IMA J Numer Anal, 2019, 39: 2016–2044
Wazwaz A M. New travelling wave solutions to the Boussinesq and the Klein-Gordon equations. Commun Nonlinear Sci Numer Simul, 2008, 13: 889–901
Wu X, Wang B, Shi W. Efficient energy preserving integrators for oscillatory Hamiltonian systems. J Comput Phys, 2013, 235: 587–605
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11771162, 11771128, 11871106, 11871092 and 11926356) and National Safety Administration Fund (Grant No. U1930402).
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Cao, W., Li, D. & Zhang, Z. Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations. Sci. China Math. 65, 1731–1748 (2022). https://doi.org/10.1007/s11425-020-1857-5
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DOI: https://doi.org/10.1007/s11425-020-1857-5