A new implicit energy conservative difference scheme with fourth-order accuracy for the generalized Rosenau–Kawahara-RLW equation

Dai, Weizhong

doi:10.1007/s40314-018-0685-4

Cited by 14 publications

(10 citation statements)

References 55 publications

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“…using the implicit fourth-order conservative finite-difference method developed in [127]. The first and second spatial derivatives of strain are set to 0 at the boundaries.…”

Section: K Numerical Simulationsmentioning

confidence: 99%

Transition fronts and their universality classes

2022

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Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

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“…using the implicit fourth-order conservative finite-difference method developed in [127]. The first and second spatial derivatives of strain are set to 0 at the boundaries.…”

Section: K Numerical Simulationsmentioning

confidence: 99%

Transition fronts and their universality classes

2022

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“…(1.1) have been proposed and analyzed -cf. Refs [1,2,22,28,31,42,43,[45][46][47]. However, to the best of our knowledge, all of existing momentum-preserving schemes have at most second-order accuracy in time.…”

Section: Introductionmentioning

confidence: 99%

Arbitrary High-Order Structure-Preserving Schemes for Generalized Rosenau-Type Equations

Jiang¹,

Xu²,

Zheng³

2023

EAJAM

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Arbitrary high-order numerical schemes conserving the momentum and energy of the generalized Rosenau-type equation are studied. Derivation of momentumpreserving schemes is made within the symplectic Runge-Kutta method coupled with the standard Fourier pseudo-spectral method in space. Combining quadratic auxiliary variable approach, symplectic Runge-Kutta method, and standard Fourier pseudo-spectral method, we introduce a class of high-order mass-and energy-preserving schemes for the Rosenau equation. Various numerical tests illustrate the performance of the proposed schemes.

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“…Later, He [32] derived the exact solitary wave solution with power law nonlinearity and advanced a three-level linearly implicit difference approach. Wang and Dai [33] developed a three-level conservative fourth-order FD approach, while Gazi et al [34] employed a septic B-spline collocation finite element (FE) technique.…”

Section: Introductionmentioning

confidence: 99%

Solitary Wave Propagation of the Generalized Rosenau–Kawahara–RLW Equation in Shallow Water Theory with Surface Tension

AL-saedi,

Nikan,

Avazzadeh

et al. 2023

Symmetry

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This paper addresses a numerical approach for computing the solitary wave solutions of the generalized Rosenau–Kawahara–RLW model established by coupling the generalized Rosenau–Kawahara and Rosenau–RLW equations. The solution of this model is accomplished by using the finite difference approach and the upwind local radial basis functions-finite difference. Firstly, the PDE is transformed into a nonlinear ODEs system by means of the radial kernels. Secondly, a high-order ODE solver is implemented for discretizing the system of nonlinear ODEs. The main advantage of this technique is its lack of need for linearization. The global collocation techniques impose a significant computational cost, which arises from calculating the dense system of algebraic equations. The proposed technique estimates differential operators on every stencil. As a result, it produces sparse differentiation matrices and reduces the computational burden. Numerical experiments indicate that the method is precise and efficient for long-time simulation.

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A new implicit energy conservative difference scheme with fourth-order accuracy for the generalized Rosenau–Kawahara-RLW equation

Cited by 14 publications

References 55 publications

Transition fronts and their universality classes

Transition fronts and their universality classes

Arbitrary High-Order Structure-Preserving Schemes for Generalized Rosenau-Type Equations

Solitary Wave Propagation of the Generalized Rosenau–Kawahara–RLW Equation in Shallow Water Theory with Surface Tension

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