PhD thesis in English

first part of the propagation according to Eq. (A.6), while after the additional stepaccording to Eq. (A.7), the propagation to t+ǫ is finished, and a new wave functionφ n+1 (r) is calculated. Finally, we rewrite Eqs. (A.6) and (A.7) in the approximatediscretized form:φ n+1/2 (r) = e −iε „12 r2 +g˛ φn (r) ˛r˛2« φ n (r), (A.8)i 1 (φ n+1 (r) − φ n+1/2 (r) ) = − 1 ∂ 2 φ n+1/2− 1 ∂ 2 φ n+1. (A.9)ε4 ∂r 2 4 ∂r 2In the last equation, we have used the semi-implicit Crank-Nicolson method [137],which is highly accurate, robust and inexpensive method for solving general diffusionequations. The accuracy of the approximation is ε 2 .To reduce the obtained differential equations to the algebraic form, we additionallyperform space discretization with the discretization step h. To this end,we introduce index i, which takes values from 0 to i max − 1, and approximate thesecond-order spatial derivative in the standard way:∂ 2 φ n→ 1 ( )φn∂r 2 4h 2 i+1 − 2φ n i + φ n i−1 , (A.10)with the error of the order of h 2 . As a result, we obtain a tridiagonal system ofequations:where−Aφ n+1i+1 + Bφn+1 i − Aφ n+1i−1 = δ i ,(A.11)δ i = (ε/4h 2 )φ n+1/2i+1 + ( 1 − ε/2h 2) φ n+1/2i +(ε/4h 2 )φ n+1/2i−1 , A = ε/4h 2 , B = 1+ε/2h 2 .A solution of a tridiagonal system of equations can be cast in the form:φ n+1i+1 = α iφ n+1i + β i , (A.12)and from this Ansatz we find the recursive relations for the solution:α i−1 =AB − Aα i,β i−1 = δ i + Aβ iB − Aα i.(A.13)From the boundary condition φ n i max= φ n+1/2i max, we derive initial values for α and β:α imax−1 = 0 and β imax−1 = φ n+1/2i max, which we use to solve the recursive equations139