The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations

“…Finally, as we know, much attention has been paid on energy-preserving numerical methods for solving single or coupled wave equations (Cao et al 2017;Deng and Liang 2020;Deng and Wu 2021;Li and Sun 2020). However, as far as we know, there are no researches on numerical solutions of nonlinear wave equations by energy-conserving Richardson extrapolation methods.…”

“…Consider the following system of the coupled KG equations(Deng and Liang 2020;Deng and Wu 2021)u tt − a 2 Δu + a 1 u + b 1 u 3 + c 1 uv 2 = 0, (x, t) ∈ Ω × [0, T ], v tt − a 2 Δv + a 2 v + b 2 v 3 + c 2 u 2 v = 0, (x, t) ∈ Ω × [0, T ], on Ω × [0, T ], where Ω = [0, 1] × [0, 1]. Their exact solutions are u(x, t) = a 3 sech[ (x + y − γ t)], v(x, t) = a 4 sech[ (x + y − γ t)],…”

“…Set Ω = [−15, 15] × [−15, 15], we consider the following IBVPs (Deng and Liang 2020; Deng and Wu 2021)u tt − Δu = −δ 2 sin(u − v), (x, t) ∈ Ω × [0, T ], v tt − b 2 Δv = sin(u − v), (x, t) ∈ Ω × [0, T ], u(x, t), v(x, t) = (0, 0), (x, t) ∈ ∂Ω × [0, T ], u(x, 0), v(x, 0) = 4φ(x), φ(x) , u t (x, 0), v t (x, 0) = (0, 0), x ∈ Ω, in which φ(x) = arctan[exp(3 − x 2 + y 2 )]. Set u t = ω, v t = ϑ,the energy of this problem(Deng and Liang 2020;Deng and Wu 2021), which is defined byE(t) = 1 2 Ω αω 2 + β|∇u| 2 + γ ϑ 2 + |∇v| 2 + 2 G(u, v) dxdy, (5.1) is conservative. Here, G(u, v) = 1 − cos (u − v), α = β = δ −2 , γ = 1 and = b 2 .…”

Fully decoupled, linear and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

“…The classic coupled Gordon-type systems have been a hot topic of detailed investigations in nonlinear science, and widely applied in many fields, such as but not limited to the plasma physics [6], the flux propagation in Josephson junctions between two superconductors [28], the laser pulses [20], vibrations of DNA molecules [36], the propagation of an optical pulse in fibre waveguide and the long-wave dynamics of two coupled periodic chains of particles [1,22] etc. Moreover, there also exist some excellent works on the exact solutions [10,17,29] and numerical solutions [3,4,9,15,23,39] for the coupled Gordon-type equations.…”

“…Based on the above-mentioned auxiliary variable methods, Deng and Wu [9] constructed an energy-preserving numerical scheme by employing the IEQ method and the Crank-Nicolson method for the classic sine-Gordon equation and obtained the convergence analysis, although the proposed scheme is linearly implicit, the algebraic equation is still coupled and equips with the variable coefficient matrix. More recently, Guo and Mei et.…”