Igor V. Puzynin

Finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations

Abstract The finite element method (FEM) is applied to solve the bound state (Sturm-Liouville) problem for systems of ordinary linear second-order differential equations. The convergence, accuracy and the range of applicability of the high-order FEM approximations (up to tenth order) are studied systematically on the basis of numerical experiments for a wide set of quantum-mechanical problems. The analytical and tabular forms of giving the coefficients of differential equations are considered. The Dirichlet and Neumann boundary conditions are discussed. It is shown that the use of the FEM high-order accuracy approximations considerably increases the accuracy of the FE solutions with substantial reduction of the requirements on the computational resources. The results of the FEM calculations for various quantum-mechanical problems dealing with different types of potentials used in atomic and molecular calculations (including the hydrogen atom in a homogeneous magnetic field) are shown to be well converged and highly accurate.

A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

Abstract A generalization of the Crank–Nicolson algorithm to higher orders for the time-dependent Schrodinger equation is proposed to improve the accuracy of the time approximation. The implicit difference schemes are obtained in terms of the Magnus expansion for the evolution operator and its further factorization with the help of diagonal Pade approximations. Stability of the schemes and conservation of the approximated solution norm are provided by the fact that the Magnus expansion of the evolution operator preserves its unitarity in any order with respect to a time step τ . As an example, a comparison between the numerical and analytical solutions of a model problem for the oscillator with an explicitly time-dependent frequency was performed for the schemes O ( τ 4 ) and O ( τ 6 ) to demonstrate accuracy, efficiency and adequate convergence of the method.

FESSDE, a program for the finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations

Abstract A FORTRAN 77 program is presented which solves the Sturm-Liouville problem for a system of coupled second-order differential equations by the finite element method using high-order accuracy approximations. The analytical and tabular forms of giving the coefficients of differential equations are considered. Zero-value (Dirichlet) and zero-gradient (Neumann) boundary conditions are also considered.

Magnus-factorized method for numerical solving the time-dependent Schrödinger equation

Abstract The method of constructing the high-order stable operator-difference schemes for solving the time-dependent Schrodinger equation are proposed as a generalization of the Crank–Nicolson scheme. The schemes are constructed in terms of the Magnus expansion for the evolution operator. The case of the non-homogeneous equation is also considered.

Newtonian iterative scheme with simultaneous iterations of inverse derivative

Abstract A modification of the Continuous Analogy of Newton Method for the numerical solving of nonlinear problems is suggested. It permits one to replace the inversion of the derivative operator on every step of iterations by its inversion only in the initial approximation point. Then the extended system of the differential equations in Hilbert space, introduced in the work, permits the realization of the iterative process with the simultaneous calculation of the inverse derivative operator. The convergence theorem is proved for the method under almost the same conditions as for CANM. Numerical calculations for the model problem (Kirchhoff equation) have shown the effectiveness and adequately fast convergence of the iterative schemes based on the suggested method.

A Hybrid Numerical Method for Analysis of Dynamics of the Classical Hamiltonian Systems

Abstract The numerical integration methods based on the forward and backward expansions of solutions in the Taylor series for some classical Hamiltonian systems are considered. The analytical representations of derivatives of the Hamiltonian are used for construction of the hybrid schemes of the approximate solutions of the Cauchy problem. The considered approach allows us also to study the solutions in the neighborhood of the singular points of the Hamiltonian. The efficiency of these hybrid implicit methods is illustrated on examples of numerical analysis of solutions for some Hamiltonian systems such as Toda and Henon-Heiles models, the system of Coulomb particles, and the three-body gravitational system on a line. A discrete time representation of the evolution of the three-body system on a line connected with constructing pair collision transition operators and Poincare sections is discussed.

Numerical simulations of heat and moisture transfer subject to the phase transition

A model for the description of the heat and moisture transfer in a porous material is proposed. The density of the saturated vapor and the transfer coefficients of the liquid and vapor moistures depend on the temperature. At the same time, the conductivity coefficient of the porous material depends on the moisture. On the basis of the proposed model, a numerical simulation of the heat and moisture transfer for a drying process has been performed.

FESSDE 2.2: A new version of a program for the finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations

A FORTRAN program is presented which solves the Sturm-Liouville problem for a system of coupled second-order differential equations by the finite element method using high-order accuracy approximations. The analytical and tabular forms of giving the coefficients of differential equations are considered. Zero-value (Dirichlet) and zero-gradient (Neumann) boundary conditions are also considered.

Scheme of splitting with respect to physical processes for a model of heat and moisture transfer

A difference scheme of splitting with respect to physical processes for a model of heat and moisture transfer is proposed. The model involves three physical processes—heat, liquid and saturated vapor transfer in the porous material. The density of saturated vapor and the transfer coefficients of liquid and vapor moistures depend on the temperature. At the same time, the heat capacity and conductivity of the porous material depend on moisture. On the basis of the proposed scheme of the model, a numerical simulation of the heat and moisture transfer for a drying process has been performed.

A numerical method for determination of moisture transfer coefficient according to the diffusion moisture profiles

For a set of the measured diffusion moisture profiles, a numerical method for determination of moisture transfer coefficient D(w, t) is suggested. The transfer coefficient is found as a sum of the degree \( p_0 w^{p(t)} \) and exponential \( Ae^{\mu (w - v_0 )} \) functions of the moisture concentration w, as opposite to the previous works. The exponent p(t) of the power function depends on time t. The exponential function describes profiles for large times nearby the boundary of the sample, where the moisture evaporation takes place to the atmosphere. A conservative difference scheme for numerical solution of direct problem is suggested. An inverse problem for minimization of an error functional is solved by the Newton method. Thus, a more accurate coincidence of the calculated profiles of the moisture concentration to the measured profiles is gained.