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In this paper, two efficient high-order physical property-preserving linearly implicit finite difference schemes are firstly developed and analyzed for nonlinear fourth-order wave equation with damping based on the scalar auxiliary variable (SAV) approach and newly developed Lagrange multiplier (LM) approach, respectively. Then the modified energy preservation property, the priori bounds of the numerical solution and convergence analysis of the former scheme are presented in detail. It is pointed out that the former scheme only conserves/dissipates the modified discrete energy, not the original energy while the latter scheme conserves/dissipates the original discrete energy exactly. Also, the latter scheme only demands to solve a decoupled and linear system with constant coefficients at each time step plus a nonlinear algebraic system. Finally, some numerical simulations are presented to demonstrate the accuracy, efficiency and preservation property of the obtained schemes. • Two efficient physical property-preserving difference methods for the nonlinear fourth-order wave equation with damping are proposed. • The modified preservation property and error estimation of the former scheme are presented in detail. • Both schemes demand to solve a sequence of linear decoupled systems with constant coefficients at each time step. • Some numerical results are performed to demonstrate the accuracy, efficiency and preservation property of both schemes. [ABSTRACT FROM AUTHOR]

This paper concentrates on the error estimation of novel time second‐order splitting conservative finite difference method (FDM) for high‐dimensional nonlinear fractional Schrödinger equation rigorously. The discrete preservation property of our scheme is exhibited. By virtue of the cut‐off technique and discrete energy method, it is shown that our scheme possesses the accuracy of O(Δt2+hx2+hy2) in sense of L2‐ norm. Numerical experiments are exhibited to validate the accuracy and conservation property of our scheme. [ABSTRACT FROM AUTHOR]

• It is difficult to develop efficient conservative difference schemes for space fractional Klein-Gordon-Zakharov system. • New auxiliary equations are introduced to transform the original fractional system into the equivalent system of equations. • Two novel linearly implicit conservative difference schemes for the transformed system are developed and analyzed. • The discrete energy conservation properties, unconditional convergence of the suggested schemes are proved rigidly. • Numerical results are exhibited to study the physical behavior of the wave propagation and confirm efficiency of schemes. In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂ u ∂ t = − v and ∂ ϕ ∂ t = ∂ 2 ψ ∂ x 2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O (Δ t 2 + h 2) and O (Δ t 2 + h 4) in L ∞ - and L 2 -norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes. [ABSTRACT FROM AUTHOR]

• Unconditionally convergent energy dissipative numerical methods are developed and analyzed for nonlinear wave equations. • Two novel efficient energy dissipative difference schemes for the wave equations are first set forth and analyzed. • The discrete energy dissipation properties, solvability, unconditional convergence and stability results are proven rigidly. • Some numerical results are provided to illustrate the computational accuracy and efficiency of the proposed schemes. In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional convergence and stability of the proposed schemes are exhibited rigidly. By the discrete energy analysis method, it is rigidly shown that the proposed schemes achieve the unconditional convergence rates of O (Δ t 2 + h 2) in the discrete L ∞ -norm for the associated numerical solutions. At last, some numerical results are provided to illustrate the dynamical behaviors of the damping terms and unconditional energy stability of the suggested schemes, and testify the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]

• The existence of energy invariant for the fractional BE is derived. • An energy-preserving splitting CN scheme for the fractional BE is established. • A new potential function is introduced to ensure the conservation of energy. • Discrete energy property, convergence and unconditional stability are proved. In this paper, an efficient energy-preserving splitting Crank-Nicolson difference scheme for the fractional-in-space Boussinesq equation is established. The novelty of the proposed scheme here is that the potential function v is introduced via ∂ u ∂ t = − (− Δ) α 2 v to ensure the conservation of energy. By utilizing the discrete energy method, it is shown that the proposed scheme attains the convergence rates of O (Δ t 2 + h 2) in the discrete L ∞ -norm without any restrictions on the grid ratio, and is unconditionally stable. Finally, a linearized iterative algorithm is proposed and numerical results are provided to demonstrate the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient physical-property-preserving algorithm for the space fractional-order generalized Zakharov system is proposed and analyzed. Firstly, the space fractional-order generalized Zakharov system is reformulated as an equivalent system of equations by introducing the auxiliary equation. Then the spatial fourth-order physical-property-preserving linearly implicit difference scheme is developed for the transformed system. Subsequently, with the aid of the cut-off function and discrete energy analysis method, the underlying scheme are proven to be with the optimal order of O (Δ t 2 + h 4) in discrete L ∞ and L 2 norms. The main feature of the new scheme is physical-property-preserving, linearly decoupled and easy to be applied in parallel computing, especially in long time simulations and large-scale problems. At last, ample numerical results are exhibited to substantiate the efficiency and preservation properties of our scheme, and investigate the dynamic behaviors of the collision of different solitary waves with subsonic and supersonic propagation speeds, respectively. • An efficient physical-property-preserving difference scheme for space fractional generalized Zakharov system is proposed and analyzed. • The main feature of the new scheme is physical-property-preserving, linearly decoupled and easy to be applied in parallel computing. • The discrete conservation properties and optimal error estimation of the proposed scheme are proven. • Some numerical results are presented to simulate the collision of different solitary waves with subsonic and supersonic propagation speeds. [ABSTRACT FROM AUTHOR]