A Space-Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

On the basis of sampling approximation for a function defined on a bounded interval by combining Coiflet-type wavelet expansion and technique of boundary extension, a space-time fully decoupled formulation is proposed to solvemultidimensional Schrödinger equations with generalized nonlinearities and damping. By applying a wavelet Galerkin approach for spatial discretization, nonlinear Schrödinger equations are first transformed into a system of ordinary differential equations, in which all matrices are completely independent of time and never need to be recalculated in the time integration. Then, the classical fourth-order explicit Runge–Kutta method is used to solve the resulting semidiscretization system. By studying several widely considered test problems, results demonstrate that when a relatively fine mesh is adopted, the present wavelet algorithm has a much better computational accuracy and efficiency than many existing numerical methods, due to its higher order of convergence in space which can go up to 6.

Because of the broad applications of GNLSE (1), developing accurate and efficient numerical methods for solving such equations has attracted considerable research attention in the past few years.For instance, Mohebbi and Dehghan [15] studied two-dimensional linear Schrödinger equations by using a compact boundary value method with fourth-order accuracy in both space and time.Liao et al. [16] applied a fourth-order compact difference scheme to solve the two-dimensional linear Schrödinger equation without damping term.In order to save cost on computation, the finite difference schemes combining with the alternating direction implicit method are developed to study the multidimensional Schrödinger equations [18-20, 23, 24, 30, 34].In such algorithms, handling multidimensional problems is transformed into solving a series of one-dimensional problems by introducing intermediate variables.Moreover, many other numerical methods are also proposed to solve multidimensional Schrödinger equations, such as the collocation method [21,22,27,33], the Galerkin method [27,31], and the mesh-free methods [25,31,32].The above methods are effective for solving Schrödinger equations under certain conditions.However, many of them will encounter severe difficulties in uniformly solving the three-dimensional generalized nonlinear Schrödinger equation (1).For example, the time-splitting methods need to obtain the density  by solving analytically a nonlinear differential equation [7,8,21,26], which is an extremely difficult task for general damping term ().And when the classical collocation method [25] and Galerkin type method [32] are employed to solve directly the nonlinear Schrödinger equation, the matrices generated in the spatial discretization of nonlinear terms will be dependent on the time-dependent unknown vector and must be recalculated at each time step [25,32,38], thereby consuming considerable computing resources.Because such repeated recalculations of matrices from the spatial discretization of nonlinear term can be regarded as reperforming the spatial discretization at each time step [38], these methods [25,27,32] cannot divide the solution procedure into two completely separate processes, that is, the spatial discretization and the time integration.Therefore, the decoupling between spatial and temporal discretizations in these methods [25,27,32] is incomplete [38].
In the current work, a space-time fully decoupled formulation by combining a wavelet Galerkin technique and the classical fourth-order explicit Runge-Kutta method is proposed to uniformly solve the three-dimensional generalized nonlinear Schrödinger equation (1).In such a space-time fully decoupled wavelet formulation, all matrices generated in the spatial discretization of the nonlinear partial differential equation ( 1) are constant matrices and need not to be updated in the subsequent time integration.In addition, a systematic comparison between the present solutions and those obtained by using many existing numerical methods is conducted by solving several widely considered test problems.

Solution of the Generalized Nonlinear Schrödinger Equation
At the beginning of the solution to the three-dimensional generalized nonlinear Schrödinger equation ( 1), by introducing (x, ) =  1 (x, ) + i 2 (x, ), where  1 (x, ) and  2 (x, ) are real-valued functions, we rewrite it into two coupled timedependent nonlinear partial differential equations in which  = 1, 2 and  = 3 − .
Finally, one can obtain an approximate solution of GNSE (1) by employing a time integration scheme to solve the nonlinear ordinary differential equations (15).In this study, the classical fourth-order explicit Runge-Kutta method is adopted, and then we have in which   = Δ, and Δ is the time step.Iteratively applying (19) and directly using the initial condition (2), one can obtain the unknown vector U(  ) = {U 1 (  ), U 2 (  )} T at each time step, which can be used to reconstruct the approximate solution (x, ) based on scheme (10).

Numerical Examples
In this section, we use several numerical examples to demonstrate the accuracy and applicability of the proposed wavelet method for solving two-and three-dimensional generalized nonlinear Schrödinger equations (1).
To effectively evaluate the performance of numerical solutions, we introduce the error norms  max and  2 , and the HOC-ADI-I [18] HOC-ADI-II [18] PR-ADI [6,18] SD-HOC [10,18] L-HOC-ADI [24] Present Number of grid points N s 10 1  10 2 Error norm corresponding convergence rates in space  max and  2 , which are, respectively, defined by [9,18,20,23] in which  num and  exact are the numerical and exact solutions, respectively, and   = 2  is the number of grid points in space.To make the convergence rates  max and  2 estimated by ( 22) being spatial convergence rates, a time step Δ small enough to guarantee that the errors mainly come from the spatial discretization is adopted in the following computation.
Example 2. We consider the two-dimensional cubic nonlinear Schrödinger equation [34] i subjected to the initial and boundary conditions extracted from the exact solution (, , ) = exp[i( +  − )].In Table 1, we list the error norms  2 of numerical solutions at time  = 1 under Δ = 1/25000 achieved, respectively, from the proposed wavelet method, and the higher order compact alternating direction implicit method (HOC-ADI) [34].
with the initial condition and boundary condition extracted from the exact solution (, , ) = sin(+)exp(−i).Table 3 shows the spatial convergence of the present wavelet solutions at time  = 1 and 3. Error norm Example 7. We consider the three-dimensional nonlinear Schrödinger equation [9].
It can be seen from Tables 1-4 and Figures 2-4 that the proposed space-time fully decoupled wavelet Galerkin method has a good accuracy, efficiency, and stability.Moreover, these results also demonstrate that its order of convergence in space can go up to 6.In addition, we can see from the comparisons shown in Figures 2-4 and Tables 1-4 that when a relatively fine mesh is adopted, the present wavelet solutions have a much better numerical accuracy than those obtained by many other existing numerical methods, including PR-ADI [6,18], SD-HOC [10,18], HOC-ADI [18,23,34], L-HOC-ADI [24], FONCECD [28], FOLCECD [28], SSFD [9], and L-ADI [24].

Conclusion
In this study, we introduced a sampling approximation for a function defined on a three-dimensional domain by combining Coiflet-type wavelet expansion and technique of boundary extension.On the basis of such a wavelet approximation, a space-time fully decoupled wavelet Galerkin is proposed to solve uniformly the multidimensional generalized nonlinear Schrödinger equation with damping, in which all matrices generated from the spatial discretization of nonlinear partial differential equations are completely independent of time and need not be updated in the time integration.By solving seven widely considered test problems, the present wavelet algorithm shows considerably high precision and fast convergence rate in space compared with many other existing numerical methods.

Figure 1 :
Figure 1: The sparsity pattern of matrix A for  = 6.
The error norms of the present wavelet solutions at time  = 1 and 3 under various space-time meshes are listed in Table2.

Table 2 :
Error norms of wavelet solutions at  = 1 and 3 with Δ = 0.001 for Example 5.

Table 3 :
Error norms of wavelet solutions at  = 1 and 3 with Δ = 0.0005 for Example 6.

Table 4 :
Error norm  max and convergence rate  max of numerical solutions at  = 2 for Example 7.