Algorithmic and software support
of theoretical investigations

Computational molecular dynamics methods and software. An optimized version of the DL_POLY molecular dynamics simulation code has been used to study the cluster-surface impact processes for metallic phases. The characteristics of the cluster-surface collisions were studied in a wide range of the cluster impact energies (Einc = 0.035-3.5 eV/atom). Modification of the surface, exposed to the cluster beams, was studied by monitoring the molecular dynamics configurations of the system in real time. The density and temperature distributions in the system under the energetic irradiation have been investigated in detail. The three major channels of the impact yield (viz., soft landing, droplet spreading and implantation) were distinguished and estimated. Based on the density and temperature distributions data the low-energy cluster-surface impact has been analyzed and a novel interpretation of droplet spreading process given [14]. Fig. 6 shows the top (a), side (b) and cut (c) views of MD configurations at t = 1.3, 2.1 and 5.0 ps for the cluster incident energy Einc = 0.56 eV/atom.

    Fig. 6. Results of MD simulation for cluster. The top (a), side (b) and cut (c) views of MD configurations at t = 1.3, 2.1 and 5.0 ps for cluster incident energy Einc = 0.56 eV/atom.

Thermoelastic computational approach to beam-surface interaction modeling. A method of numerical analysis of Stefan's problem for a metal sample exposed to a high-current pulse ion beam has been developed [15]. In supposition that the lateral areas of the sample are thermoisolated, the dynamics of moving the interphase separating the melted and firm parts of the sample was investigated. It has been found that the form of the source influences the form of the interphase. Therefore, choosing the characteristics of the source, one can control the evolution of the interphase.

High level accuracy computational schemes for quantum systems investigations. An uncoupled correlated variational method for the calculation of helium isoelectronic bound states has been proposed. New projective coordinates s = r1+r2, v = r12/(r1+r2), w = (r1-r2)/r12 are introduced instead of the conventional ones s = r1+r2, t = r1-r2, u = r12. All matrix elements of the total Hamiltonian and the weight function are expressed as simple products of three one-dimensional integrals. The variational basis is formed by a set of Laguerre polynomials with a single nonlinear parameter and two sets of Jacobi polynomials for the projective coordinates s, v, w respectively. It provides a reasonable rate of convergence of the energy, = (N) with respect to a number N of the basis components of the eigenvector. The proposed method yields the best available energies for the isoelectronic ground states of the helium atom. New estimations of the isotope helium ground states were also presented [16].

A Newtonian iteration scheme has been constructed in framework of research on computational physics for solving a scattering problem using the Schwinger variational functional. The scattering problem is formulated as an eigenvalue problem with respect to a pair of unknowns: a phase shift and a wave function. The efficiency of the proposed iteration scheme and its accuracy are demonstrated by exact solvable examples of the elastic scattering problem with Morze and spherical potentials [17].

Numerical calculations for nuclear models. The problem of production and survival of the very-long-lived isomeric state of 180 nuclide is a real challenge for a theory of nucleosynthesis. Numerical calculations were performed within the Quasiparticle-Phonon Nuclear Model (QPNM) using the formalism developed in works of V.G.Soloviev. The calculations performed allowed one to make a conclusion about mechanisms of transitions of intermediate states of de-excitation of 180m in the reaction [18].

Computer algebra. The original highly efficient algorithms for computation of Janet bases were designed and implemented in Reduce, C and C++. These algorithms exploit a very useful data structure called Janet tree for representing the multiplicative variables with respect to the Janet division of a given monomial set. Based on this data structure the completion to involution of both monomial and polynomial sets can be performed extremely fast. The new algorithms admit the further optimization. Extensive benchmarking was performed in comparison with the special-purpose computer algebra system SINGULAR. This is a system dedicated to polynomial computations and is considered as very fast for Groebner basis computations. In most examples, the implementations of the new algorithms turned out to be faster. Moreover, the new algorithms unlike classical Buchberger algorithm for computation Groebner bases admit effective parallelization, what was explicitly demonstrated on a two-processor Pentium based computer. Modeling multi-processor computations on this machine reveals a behaviour of the computing time which is close to inversely proportional with respect to the number of processors. General involutive algorithms for polynomial as well as linear systems of partial differential equations were also implemented in Mathematica [19-21].

Computation of cohomologies for Lie algebras and superalgebras explicit computation of Lie (super)algebra cohomologies is of great importance for studying modern models of theoretical and mathematical physics. A new algorithm has been designed. The new algorithm splits cochain complexes containing spaces of very high dimension into smaller ones. In many applications this strategy leads to significantly faster computations. The algorithm has been implemented in C and applied to some concrete examples of physical interest. This approach can also be applied to explicitly determine the Spencer cohomology of Z-graded Lie (super)algebras [22].

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