Implicit spectral methods for wave propagation problems

Wineberg, Stephan B; Mcgrath, Joseph F.; Gabl, E. F.; Scott, L. Ridgway; Southwell, Charles E

doi:10.1016/0021-9991(91)90200-5

Cited by 7 publications

(9 citation statements)

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“…The second statement then easily follows, by taking into account (80), from the fact that, see (6), (74), (75), and (78):…”

Section: Fourier Space Discretizationmentioning

confidence: 98%

“…In the method of lines approach, the spatial derivatives are usually approximated by finite differences or by discrete Fourier transform and the resulting system is then integrated in time by a suitable ODE integrator. Spectral methods have revealed very good potentialities especially in the case of periodic boundary conditions [39,75]. 1 For weakly nonlinear term f in (1), the modulated Fourier expansion technique [47, Chapter XIII] has been adapted to both the semi-discretized and the full-discretized systems to state long-time near conservation of energy, momentum, and actions [48,33].…”

Section: Introductionmentioning

confidence: 99%

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Energy conservation issues in the numerical solution of the semilinear wave equation

Brugnano

Frasca-Caccia

Iavernaro

2015

Applied Mathematics and Computation

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In this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the prescribed boundary conditions. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schrödinger equation.

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“…The second statement then easily follows, by taking into account (80), from the fact that, see (6), (74), (75), and (78):…”

Section: Fourier Space Discretizationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Energy conservation issues in the numerical solution of the semilinear wave equation

Brugnano

Frasca-Caccia

Iavernaro

2015

Applied Mathematics and Computation

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“…In particular, when simulating soliton solutions one must take into account that line solitons are not localized objects, but they extend through the boundaries of any finite computational window. The approach we used here is based on the one in [30–34], but with different boundary conditions. For the x ‐direction, we set our computational window to be wide enough that any initial solitary waves are far away from boundary.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Soliton Interactions of the Kadomtsev–Petviashvili Equation and Generation of Large‐Amplitude Water Waves

Biondini

Maruno²,

Oikawa

et al. 2009

Stud Appl Math

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We study the maximum wave amplitude produced by line-soliton interactions of the Kadomtsev-Petviashvili II (KPII) equation, and we discuss a mechanism of generation of large amplitude shallow water waves by multi-soliton interactions of KPII. We also describe a method to predict the possible maximum wave amplitude from asymptotic data. Finally, we report on numerical simulations of multi-soliton complexes of the KPII equation which verify the robustness of all types of soliton interactions and web-like structure.

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“…Yan proposed three conservative finite volume element schemes based on the discrete variational derivative method [26], but there was no analysis of convergence. Winebery developed an implicit-stepping scheme for KdV equation in temporal direction and spectral methods in space [25]. However, there was a restriction on the size of the time step when they applied predictor-corrector method to retain the full accuracy of the scheme.…”

Section: Introductionmentioning

confidence: 99%

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

sci¹

2020

NMTMA

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Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a threelevel linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.

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Implicit spectral methods for wave propagation problems

Cited by 7 publications

References 0 publications

Energy conservation issues in the numerical solution of the semilinear wave equation

Energy conservation issues in the numerical solution of the semilinear wave equation

Soliton Interactions of the Kadomtsev–Petviashvili Equation and Generation of Large‐Amplitude Water Waves

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

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