In this thesis, we develop the CESE method in momentum space. We investigate the basic one-dimensional wave equation, convection equation, convection-diffusion equation, nonlinear Korteweg-de Vries equation, shock wave Burgers’s equation and a quantum mechanical problem.
The scope is on the fundamental part. In each problem, the momentum space CESE core scheme is developed for an explicit time-marching scheme. It is straightforward for linear problems.
While for nonlinear problem such as KdV equation and Burgers’s equation the convolution integral of the unknown functions in the source term is involved. We employ the half-step grid size for the convolutions and the iterations in each half-time marching step for the nonlinearity.
Above method we has introduced is sufficient only to uniform grid problem. For non-uniform grid problem, it is not solved. We finally figure out the idea of the solution element must be introduced as suitable modification. The troubles occur in convolution integral for nonlinear problems is conquered by a consistent treatment. When numerical grid is not on the node points, it is just expanded as a Taylor expansion from the nearly solution element. For the non-reflecting boundary condition, the zero boundary condition and the ghost cell method is employed. It shows that the ghost cell method is excellent in coordinate and momentum space, even when the domain size is small. Because of the boundary value is sufficiently small in momentum space. It is enough to use the zero boundary condition. For non-uniform grid momentum space CESE method, we introduce a transformation method to shorter the computational time and improve the convolution integral accuracy. The importance of this approach for higher-dimensional problems will be investigated in our future studies. We calibrate each system with known exact solution. We have shown that the momentum space CESE method works well for the systems from classical wave equation, nonlinear equation to quantum mechanical problem. And the error behavior of the developed
scheme is second-order. Strictly speaking, the method used in this work is not regular core scheme. But it is still capable of generating highly accurate solution by using only the concept of flux conservation and simple approximation techniques. The main advantages for the momentum space CESE method, compared to the traditional CESE method in coordinate space are twofold.
First, the boundary conditions are fulfilled automatically. That is, for sufficient large momentum value, the function and its derivatives are simply vanishing at the numerical boundary. This is because the kinetic energy of a system is physically finite. Second, the information of the wave is preserved completely inside the numerical momentum region without flowing out from the boundary like the coordinate space method. This will be especially useful in treating scattering problems in the future.With the efficient momentum space CESE method, we are able to calculate the ATI photoelectron spectra. We elucidate the capability of this method with the atom either under a very high intensity or a very long duration laser pulse to show the nice features. There is no loss of the continuous part of the wave functions, unlike filtering function employed in coordinate space method to prevent boundary reflection. In this simulation, the conspicuous enhancements appear in the high-energy part of the above-threshold (ATI) spectra, too. Because no information of wave is lost. Comparing with experiment data, the high-energy part ATI spectra obtained from our method is more accurate than the method from other coordinate space method. Further applications of the method to intense laser pulses on atoms and molecules will be presented in the future. In the future, we will develop the higher order of accuracy, and the higher dimensional momentum space CESE method to practical useful in solving realistic time-dependent problems.
References (or Bibliography)
[1]. S.C. Chang and W.M. To, "A New Numerical Framework for Solving Conservation Laws – The Method of Space-Time Conservation Element and Solution Element," NASA/TM 104495 (1991), J. Comput. Physics, 119, 295 (1995).
[2]. X.-Y. Wang, C.-Y. Chow and S.C. Chang, "Application of the Space-Time Conserva-tion Element and SoluConserva-tion Element Method to One-Dimensional AdvecConserva-tion-Diffusion Problems,"
NASA/TM 209068 (1999).
[3]. S.-C. Chang, A. Himansu, C.-Y. Loh, X.-Y. Wang and S.-T.J. Yu, "Robust and Simple Non-Reflecting Boundary Conditions for the Euler Equations, A New Approach Based on the Space-Time CE/SE Method," NASA/TMX2003-212495/REV1 (2003).
[4]. S.C. Chang and X.Y. Wang, "Multi-Dimensional Courant Number Insensitive CE/SE Euler Solvers for Applications Involving Highly Nonuniform Meshes," AIAA Paper (2003)-5285, pre-sented at the 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 20-23 (2003), Huntsville, AL.
[5]. S.C. Chang, "Courant Number and Mach Number Insensitive CE/SE Euler Solvers," AIAA Paper (2005)-4355, presented at the 41th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 11-13 (2005), Tucson, AZ.
[6]. S.C. Chang, "On Space-Time Inversion Invariance And Its Relation To Non-dissipatedness Of a CESE Core Scheme," 42th AIAA Joint Propulsion Conference (2006).
[7]. S.C. Chang, "The Space-Time CESE Method—motivating ideas, basic schemes and its recent developments," first Taiwan-USA workshop on CESE Method (2007).
[8]. More details and refferences can be found in http://www.grc.nassa.gov/WWW/microbus/.
[9]. C. Cohen-Tannoudji, J. Dupont-Roc, C. Fabre, and G. Grynberg, "Comment on the Momentum-Translation Approximation," Phys. Rev. A8, 2747 (1973).
[10]. T.F. Jiang and S.-I. Chu, "High-order harmonic generation in atomic hydrogen at 248 nm:
Dipole-moment versus acceleration spectrum," Phys. Rev. A46, 7322 (1992).
[11]. U.L. Pen and T.F. Jiang, "Strong-field effects of the one-dimensional hydrogen atom in mo-mentum space," Phys. Rev. A46, 4297 (1992).
[12]. U.L. Pen and T.F. Jiang, "Direct momentum-space calculations for the resonant multiphoton processes of a hydrogen atom under intense laser pulses," Phys. Rev. A53, 623 (1996).
[13]. Joseph Wassaf, Valerie Veniard, Richard Taieb, Alfred Maquet, "Roles of resonances and rec-ollisions in strong-field atomic phenomena: Above-threshold ionization," Phys. Rev. A 67, (2003).
[14]. Joseph Wassaf, Valerie Veniard, Richard Taieb, Alfred Maquet, "Strong Field Atomic
Ioniza-tion: Origin of High-Energy Structures in Photoelectron Spectra,"Phys. Rev. Letters (2003).
[15]. T.F. Jiang, "Calculation of atomic hydrogen and its photoelectron spectra in momentum space," Comp. Phys. Commun. 178, 571 (2008).
[16]. P.G. Drazin and R.S. Johnson, "Solitons : an Introduction," (Cambridge Univ. Press, Cam-bridge, 1989).
[17]. G.B. Arfken and H.J. Weber, "Mathematical Method for Physicists," 4th Ed. (Academic Press, San Diego, 1995).
[18]. Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe and Henk van der Vorst, edi-tors. "Templates for the Solution of Algebraic Eigenvalue Problems : A Practical Guide," SIAM, Philadelphia, (2000).
[19]. J.H. Mathews and K.D. Fink, "Numerical Methods Using Matlab," 4th Ed. (Pearson Prentice Hall, London, 2004).
Appendix A The definition of weight factor in the wiggle-suppressing scheme
The wiggle-suppressing scheme to be abbreviated as w-α scheme. The extension is formed by Eq. (2.21) and
(ux)nj = (ω−)nj (bux+)nj + (ω+)nj (bux−)nj (A.1) with
(ω±)nj = W±
³
(bux−)nj , (bux+)nj , α
´
=
¯¯¯(bux∓)nj¯¯¯α
¯¯¯(bux−)nj¯¯¯α+¯¯¯(bux+)nj¯¯¯α. (A.2) Because the scheme is an extension of the c-τ∗scheme in which (ux)nj is expressed as an weighted average of (bux−)nj and (bux+)nj . In case that¯¯¯(bux+)nj / (bux−)nj¯¯¯ is very close to 1, the only way to prevent the weighted average from becoming almost a simple average is to increase the value of α used. However, this approach is impracticable because evaluation of xα would be hamperd by very large round-off errors if α becomes very large. As such, there is a need to introduce new-weighted-averaging techniques that do not have the limitation discussed above. Two weighted-averaging formulae much more potent and flexible that discussed here were described briefly.
For motivation, note that Eqs. (A.1) and (A.2) can be expressed as
(ux)nj = ω1x1+ ω2x2 (A.3)
and
ω1 = s1
s1+ s2 and ω2 = s2
s1+ s2 (s1+ s2 > 0) (A.4)
respectively if
x1 ≡ (bux−)nj and x2 ≡ (bux+)nj (A.5) ω1 ≡ (ω−)nj and ω2 ≡ (ω+)nj
s1 ≡ ¯¯¯(bux+)nj¯¯¯α and s2 ≡¯¯¯(bux−)nj¯¯¯α (α ≥ 0)
The derivation for the first scheme is derived as following. Let δℓ≡ ωℓ− 1
2, ℓ = 1, 2. (A.6)
Thus the set{δℓ} , ℓ = 1, 2 provides a measure of how far the weighted average is deviated from the simple average. In the following, a simple way to adjust this deviation will be introduced. Let
δmin ≡ min {δℓ} and δmax≡ max {δℓ} . (A.7)
We can given any adjustable real parameter σ > 0, let
δ′ℓ ≡ σδℓ. (A.8)
For the second scheme describe as follows. To proceed, the indices of sℓ, ℓ = 1, 2, will be reshuffled such that
s2 > s1 > 0 (A.11)
Let
η1 ≡ s2
s1 − 1. (A.12)
Given any adjustable real parameter σ > 0, let (i)es1 = s1andes2 = (1 + ση1)s1 and (ii) eωℓ ≡ esℓ
Se, ℓ = 1, 2. (A.13)
where
Se≡
ÃX
ℓ
esℓ
!
> 0. (A.14)
Note that the current approach for amplifying the weight factors has one advantage over the approach described earlier, i.e., in the current approach, there is no upper bound for the value of σ one could use. Thus, in the current approach, Eq. (A.10) can be simplified as
σ = σ0
|ν| (A.15)
where σ0 > 0 again is a preset parameter in the order of 1.
Vita
My name is Zhen-Ting Huang, aka Richard. I was born on November 11, 1984. I am currently studying in National Chiao Tung University. My major is Mathematical Modeling and Scientific Computing. My research area is atomic and molecule physics and other mathematical model simulation. I excel particularly in numerical analysis, large scale matrix computing, numerical optimization (linear programming), genetic algorithm, neural network and signal processing (wavelet analysis). I am experienced in parallel computing, three-dimension graph and grid generation. In addition, I am also familiar with several kinds of mathematical software including Matlab, Mathematical and Maple.