A conservative compact difference scheme for the Zakharov equations in one space dimension

Zhou, Xuanxuan; Zhang, Luming

doi:10.1080/00207160.2017.1284319

Cited by 7 publications

(9 citation statements)

References 22 publications

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“…Firstly, we naturally derive the conservative law of the present scheme in discrete sense. Notwithstanding, the uniform boundedness of numerical solution cannot be obtained from the discrete energy law that creates technical issues when attempting to use the conventional technique like those in to analyze the proposed numerical scheme.…”

Section: Discussionmentioning

confidence: 99%

“…For this purpose, the following lemmas need to be introduced.Lemma (). On the matrices A and B, we have the following results (a) The eigenvalues of the matrices A and B are

λ_{A, j} = \frac{1}{12} (10 + 2 \cos \frac{j π}{J}), λ_{B, j} = \frac{1}{6} (4 + 2 \cos \frac{j π}{J}), j \in \overset{true^}{normalℐ} .

(b) A and B have the same eigenvectors, that is

v_{k} = (\sin \frac{k π}{J}, \sin \frac{2 italick π}{J}, …, \sin \frac{(J - 1) π}{J}), k \in \overset{true^}{normalℐ} .

(c) There are real symmetric positive define matrices R and S such that R 2 = H, S 2 = M . Lemma (). For any grid functions u ∈ X h , we have

- ⟨H δ_{x}^{2} u, u⟩ = {(‖‖, R δ_{x} u)}^{2}, ⟨M δ_{\overset{x}{true}} u, u⟩ = 0 .

…”

Section: Numerical Schemementioning

confidence: 97%

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Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Zhang

Zhou

2019

Numerical Methods Partial

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In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.

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Section: Discussionmentioning

confidence: 99%

“…For this purpose, the following lemmas need to be introduced.Lemma (). On the matrices A and B, we have the following results (a) The eigenvalues of the matrices A and B are

λ_{A, j} = \frac{1}{12} (10 + 2 \cos \frac{j π}{J}), λ_{B, j} = \frac{1}{6} (4 + 2 \cos \frac{j π}{J}), j \in \overset{true^}{normalℐ} .

(b) A and B have the same eigenvectors, that is

v_{k} = (\sin \frac{k π}{J}, \sin \frac{2 italick π}{J}, …, \sin \frac{(J - 1) π}{J}), k \in \overset{true^}{normalℐ} .

(c) There are real symmetric positive define matrices R and S such that R 2 = H, S 2 = M . Lemma (). For any grid functions u ∈ X h , we have

- ⟨H δ_{x}^{2} u, u⟩ = {(‖‖, R δ_{x} u)}^{2}, ⟨M δ_{\overset{x}{true}} u, u⟩ = 0 .

…”

Section: Numerical Schemementioning

confidence: 97%

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Zhang

Zhou

2019

Numerical Methods Partial

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“…Then, we have the following lemma. 14,27 ). Suppose that the discrete mesh function {w n |n = 1, 2, … , N; N = T} satisfies recurrence formula…”

Section: The Convergence and Stabilitymentioning

confidence: 99%

“…This article only gives the derivation of numerical methods and some numerical examples. Zhou and Zhang constructed a conservative compact difference scheme for the Zakharov equation in one space dimension and proved its convergence in order O ( τ 2 + h 4 ). For more articles about numerical methods for Zakharov equation, please see Wang, Bao and Sun, and Pan and Zhang.…”

Section: Introductionmentioning

confidence: 99%

“…[15][16][17] Among the existing numerical methods for the Zakharov equation, most of the methods and error estimates are accomplished in only one space dimension. In fact, their proofs 10,14,18,19 for difference scheme rely strongly on not only the conservation laws of the method but also the discrete version of the Sobolev inequality in one space dimension…”

Section: Introductionmentioning

confidence: 99%

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Optimal H²‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

Zhou

Wang

et al. 2019

Math Methods in App Sciences

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In this paper, a conservative compact difference scheme is proposed for the two‐dimensional nonlinear Zakharov equation with periodic boundary condition and initial condition. The proposed scheme not only conserve the mass and energy in the discrete level but also are efficient in practical experiments because the Fast Fourier transform (FFT) can be used to speed up the numerical computation. By using the standard energy method and induction argument, we can establish rigorously the unconditional and optimal H2‐error estimates. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the new scheme.

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Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

Zhang

Wang

2022

Adv Comput Math

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A conservative compact difference scheme for the Zakharov equations in one space dimension

Cited by 7 publications

References 22 publications

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Optimal H²‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

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A conservative compact difference scheme for the Zakharov equations in one space dimension

Cited by 7 publications

References 22 publications

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Optimal H2‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

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Optimal H²‐error estimates of conservative compact difference scheme for the Zakharov equation in two‐space dimension