A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation

Math Methods in App Sciences

et al. 2019

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.

“…Using the relation

E^{n} = F_{x}^{H} \overset{E^{n}}{true} F_{y}, N^{n} = F_{x}^{H} \overset{N^{n}}{true} F_{y}

yields the solution E n , N n . Note, that we can apply the Fast Fourier Transform (FFT)() to the above process.…”

Section: Algorithm and Numerical Resultsmentioning

confidence: 99%

“…Lemma For matrices M x , B x , there exist relations

M_{x} = F_{J x}^{H} {normalΛ}_{1} F_{J x}, B_{x} = F_{J x}^{H} {normalΛ}_{2} F_{J x},

where F Jx is the matrix issue from the discrete Fourier transform with elements

{(F_{J x})}_{j, k} = \frac{1}{\sqrt{J}} e^{- i \frac{2 π}{J} j k}, j = 0, 1, ⋯, J - 1, k = 0, 1, ⋯, J - 1,

F_{J x}^{H}

is the conjugate transpose matrix of F Jx , and

{normalΛ}_{1} = d i a g (λ_{M_{x}, j}), {normalΛ}_{2} = d i a g (λ_{B_{x}, j}), j = 0, 1, ⋯, J - 1 .

It is easy to extend the above results to the matrixes

M_{y}, B_{y}, H_{y} = M_{y}^{- 1}

. So we will not repeat it.…”

Section: Finite Difference Scheme and Discrete Conservation Lawmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Zhou

Math Methods in App Sciences

et al. 2019

“…So this paper can be viewed as a follow-up of our previous work in [34]. For the classical Schrödinger equation, the error estimate for the conservative Fourier spectral method has been derived recently in [35]. Moreover, the errors in the local and global energy conservation laws at the order of O( 2 ) are proved.…”

Section: Introductionmentioning

confidence: 86%

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Fei

Huang

Numerical Methods Partial

2019

This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L 2 norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results. KEYWORDSconservation law, error estimate, Fourier pseudospectral method, fractional Schrödinger equation, structure-preserving method 1 Numer Methods Partial Differential Eq. 2020;36:369-393. wileyonlinelibrary.com/journal/num

“…According to the equivalence between the semi-norms induced by the Fourier pseudo-spectral method and the finite difference method [38] and the discrete momentum conservation law, the numerical solution is proved to be bounded in the discrete L ∞ norm. We first start from an equivalent from of the RLW equation and discretize it by the Fourier pseudo-spectral method in space to arrive at a semi-discrete ordinary differential equation (ODE) system, where the momentum is conserved in the spatial semi-discrete level.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimate of two linear and momentum‐preserving Fourier pseudo‐spectral schemes for the RLW equation

Hong

Numerical Methods Partial

Gong

2019

Self Cite

In this paper, two novel linear-implicit and momentumpreserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank-Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete L ∞ norm. Then by using the standard energy method, both the linear-implicit Crank-Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of ( 2 + N −r ) in the discrete L ∞ norm without any restrictions on the grid ratio, where N is the number of nodes and is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes. KEYWORDSconservation law, error estimate, Fourier pseudo-spectral method, linear conservative scheme, momentum-preserving, regularized long-wave equationNumer Methods Partial Differential Eq. 2020;36:394-417. wileyonlinelibrary.com/journal/num