An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

Bhrawy, A. H.

doi:10.1016/j.amc.2014.08.062

Cited by 98 publications

(60 citation statements)

References 46 publications

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“…Our numerical results clearly show that ATC has better speed of convergence than NR and, with an appropriate selection of , wider radius of convergence. Applied to the complex generalized Zakharov equation with using the Chebyshev pseudo-spectral method for discretization, the ATC gives more accurate numerical solutions than they have been obtained in [18].…”

Section: Discussionmentioning

confidence: 99%

“…We will also consider several values for θ. Table 2 The results obtained in [18] are presented in Table 4. We used more number of grid points in spatial dimension than that of [18] and got better numerical accuracy in numerical results.…”

Section: Complex Generalized Zakharov Equationmentioning

confidence: 99%

“…The nonlinear complex Zakharov system has importance in plasma physics [18]. The system includes two coupled nonlinear PDEs which can be written as…”

Section: The Nonlinear Complex Generalized Zakharov Systemmentioning

confidence: 99%

“…The second test problem we consider is the GZS in complex form, which is [18]: That problem has the analytical solution…”

Section: Complex Generalized Zakharov Equationmentioning

confidence: 99%

“…In [18], two test problems concerning GZS have been solved with good accuracy by using the Jacobi pseudo-spectral collocation method. As a comparison we will solve the same two test problems with higher accuracies than those reported in [18]. The errors will be computed using the || · ||∞ norm over the entire grid as…”

Section: Numerical Analysismentioning

confidence: 99%

See 4 more Smart Citations

A parameterized multi-step Newton method for solving systems of nonlinear equations

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

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We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and, therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spacial and temporal dimensions such as, for instance, the Chebyshev pseudospectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radii of convergence than the multi-step Newton method.Keywords: Multi-step iterative methods; Multi-step Newton method; systems of nonlinear equations; partial differential equations; discretization methods for partial differential equations.

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Section: Discussionmentioning

confidence: 99%

Section: Complex Generalized Zakharov Equationmentioning

confidence: 99%

“…The nonlinear complex Zakharov system has importance in plasma physics [18]. The system includes two coupled nonlinear PDEs which can be written as…”

Section: The Nonlinear Complex Generalized Zakharov Systemmentioning

confidence: 99%

“…The second test problem we consider is the GZS in complex form, which is [18]: That problem has the analytical solution…”

Section: Complex Generalized Zakharov Equationmentioning

confidence: 99%

Section: Numerical Analysismentioning

confidence: 99%

See 3 more Smart Citations

A parameterized multi-step Newton method for solving systems of nonlinear equations

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

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The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

Parvizi

Eslahchi

2015

Math Methods in App Sciences

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In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L 2 norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced.

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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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An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

Cited by 98 publications

References 46 publications

A parameterized multi-step Newton method for solving systems of nonlinear equations

A parameterized multi-step Newton method for solving systems of nonlinear equations

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

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

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An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

Cited by 98 publications

References 46 publications

A parameterized multi-step Newton method for solving systems of nonlinear equations

A parameterized multi-step Newton method for solving systems of nonlinear equations

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

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

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