In the early 90’s, the method of space-time conservation element and solution element (to be abbreviated as CESE) was developed by Chang et al. for solving the wave problems [1][7].
Since its inception, the CESE method shows distinguished power in solving various partial differential equations (PDEs) such as problems in computational fluid dynamics, aeroacoustics, electromagnetism and magnetohydrodynamic problems etc. [8]. The concept and methodology in this method are significantly different from those in the well-established traditional method such as the finite difference, finite element, finite volume and spectral methods. The CESE method satisfies physical concept and casts the governing equation in integral form obeyed conservation law. The time marching scheme in this method is explicit. The CESE method has many nontraditional features, such as, a unified treatment of space and time, enforced both local and global flux conservation, all the dependent variables and their derivatives are considered as individual unknowns to be solved simultaneously at each grid point, the concepts of conservation element and solution element are introduced to enforce both the local and global flux conservation without using interpolation or extrapolation, and so on.
In the CESE method, the space coordinate and the time degree of freedom are treated in a unified way. The space-time domain is discretized into solution element (SE). The non-overlap space-time cells bounded by SE are called the conservation elements (CE) as depicted in Figure 2.2. The space-time flux conservation law is enforced in each CE. Time marching scheme is then derived from the conservation law. Apply the flux conservation idea at boundary CE naturally implies the non-reflecting boundary condition (NRBC) [3]. We do not need to add filter functions and absorbing potential etc. [10] near the boundary to keep the numerical solution from contaminated by the aliased reflection generated from the boundary.
However, there is a general trouble of calculation in coordinate space. Namely, we do obtain the correct information in our model numerical region, but we lose the part of wave that flows out of our space region. In some physics problems, we also interest in the wave outside of the numerical region. For examples, in the problem of highly excited states or the photoionized electron spectrum, their wave functions both extend to very large spatial range. The coordinate space calculation becomes intractable for these problems. Theoretically, we can solve a problem in either coordinate space or momentum space representation. They are equivalent and complementary to each other in case the solution is complete. Thus, a widely diffusive wave in coordinate space is transformed to narrowly localized one in momentum space. Due to extremely large energy for a system is usually unphysical. So, only a moderate momentum region is sufficient for numerical modeling. As a result, zero Dirichlet boundary condition can be imposed in the momentum space. Solving problems in momentum space is naturally attempted.
However, the application of CESE method in momentum space has never touched to our knowledge. In this article, we aim to develop a new momentum space CESE method that reserves the power of CESE and still keeps the complete information of solution simultaneously during the time evolution. A Fourier transformation can convert momentum space solution into coordinate space representation at any time if the information in the latter is requested. The momentum space approach will then be useful for both the time-dependent systems and scattering problems. This article contains the layout of the fundamental idea of the momentum space CESE method.
The purpose of investigation is to apply high accuracy numerical scheme – the "momentum space space-time conservation element and solution element (CESE) method" to simulate several one-dimensional wave equations. Several paradigmatic wave equations are solved by the method and calibrated with known solutions. Finally, we apply the method to the problem of single atom and single-active electron Schrödinger equation with strong field [11]. For particular, we
investigate the popular research topic "strong field atomic ionization" in atomic physics, such as, above-threshold ionization (ATI) spectra. We present the results for the photoelectron spectra observed when atoms are submitted to an intense laser field, and focus our discussion of the parameters’s range where conspicuous enhancements are observed in the high-energy part of the above-threshold (ATI) spectra.
Development of the CESE method in momentum space is motivated by the goal to prevent boundary reflection like method in coordinate space, and to preserve information completely for scattering states. We introduce three kind of different non-reflecting boundary conditions, and describe the complete information on how to apply it to momentum space or coordinate space method. In this article, we add an iterating process to improve numerical accuracy. For nonlinear problem, we give a modification about the convolution integral treatment which satisfies the basic idea of the solution element. In the end, we derive a non-uniform grid momentum space CESE method, and provide a method to increase accuracy about the shock wave problem. For the single-active electron systems, we introduce a simple discretization method and lead the problem into the standard eigenvalue problem. For the eigenvalue problem, we use the QZ algorithm (small size matrix) or JD method (large scale matrix) as a solver [18]. After that we obtain a pair set of eigenvalues and eigenfunctions. The reliable pseudocomplete set of momentum space eigenfunctions is then applied to the calculate of time-evolution of intense laser pulse on Ar atom.
At the end of the simulation we obtaine the final state wave function. For this state, a renormalized process is introduced to confirm the normalization.
Comparing the numerical results with the exact solutions for each case, we have shown that the momentum space CESE method has excellent results in simulating the nonlinear wave function. Moreover, the momentum space CESE method also provide a reasonable numerical solution of shock wave problem. Compare to the solution methods in coordinate space, this
method preserves the complete information of the wave during the time evolution. This is a useful feature of the momentum space method especially for the scattering state problems. With the advantage of having no boundary reflection during the time evolution, the photoelectron spectra of above-threshold ionization (ATI) are elucidated. Some of which are not feasible or very difficult to solve with the coordinate space method. Generalization of the method to single-active electron systems is straightforward. In our discussion, the conspicuous enhancements appears in the high-energy part of the above-threshold (ATI) spectra, too. Because of such the basic derivation of the momentum space CESE method is realized and verified in solving many kinds of wave equation. The developed numerical method has more potential on the following works in "strong field ionization problem". The rest of this article is organized as follows: In Sec. II, we present the formulation of the CESE method for the simple wave equation. In Sec. III, we introduce the momentum space CESE method for the simple wave equation. In Sec. IV, the nonlinear Korteweg-de Vries (KdV) equation and shock wave problem are solved by momentum space CESE method. In Sec. V, the non-uniform grid momentum space CESE method is presented. And in Sec. VI, we calculate the time-dependent Schrödinger equation of single-active electron system with an intense laser field. The numerical results are given in Sec. VII and VIII. The discussion and conclusions are given in Sec. IX.