Natalija Tumanova

On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions

AbstractIn this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy met...

Application of distributed parallel computing for dynamic visual cryptography

We consider one applied global optimization problem where a set of feasible solutions is discrete and very large. The goal is to find optimal perfect gratings, which can guarantee high quality and security of the visual cryptography method. A priori estimation techniques, such as branch and bound type methods, cannot be applied to exclude an essential part of elements from the feasible set. Thus, a full search is required to solve this global optimization problem exactly, which is very computationally demanding. A library of C++ templates is developed that allows its user to implement parallel master–slave algorithms for his/her application without any knowledge of parallel programming API (application programming interface). Design of the templates allows users to build a parallel solver using MPI (message passing interface) API or distributed computing application using BOINC (Berkeley open infrastructure for network computing) API from the same C/C++ code with implementation of application-specific tasks. We build parallel and distributed computing solvers for the considered optimization problem and present results of computational experiments using a computer cluster and BOINC project for volunteer computing. Heuristic methods are also considered as an alternative to the full search algorithm. Due to complicated conditions defining feasible solutions (perfect gratings), genetic algorithms cannot be used to solve this problem efficiently. We propose two memetic heuristic algorithms, when a basic stochastic or simplified full search algorithm is combined with a local search algorithm. Parallel heuristic algorithms are also proposed and implemented. The efficiency and accuracy of heuristics are investigated and results of experiments are presented.

Predictor-Corrector Domain Decomposition Algorithm for Parabolic Problems on Graphs

In this paper, we present a predictor-corrector type algorithm for solution of linear parabolic problems on graph structure. The graph decomposition is done by dividing some edges and therefore we get a set of problems on sub-graphs, which can be solved efficiently in parallel. The convergence analysis is done by using the energy estimates. It is proved that the proposed finite-difference scheme is unconditionally stable but the predictor step error gives only conditional approximation. In the second part of the paper it is shown that the presented algorithm can be written as Douglas type scheme, based on the domain decomposition method. For a simple case of one dimensional parabolic problem, the convergence analysis is done by using results from [P. Vabishchevich. A substracturing domain decomposition scheme for unsteady problems. Comp. Meth. Appl. Math. 11(2):241-268, 2011]. The optimality of asymptotical error estimates is investigated. Results of computational experiments are presented.

The Mathematical Modelling of Heat Transfer in Electrical Cables

Abstract This paper describes a mathematical modelling approach for heat transfer calculations in underground high voltage and middle voltage electrical power cables. First of the all typical layout of the cable in the sand or soil is described. Then numerical algorithms are targeted to the two-dimensional mathematical models of transient heat transfer. Finite Volume Method is suggested for calculations. Different strategies of nonorthogonality error elimination are considered. Acute triangles meshes were applied in two-dimensional domain to eliminate this error. Adaptive mesh is also tried. For calculations OpenFOAM open source software which uses Finite Volume Method is applied. To generate acute triangles meshes aCute library is used. The efficiency of the proposed approach is analyzed. The results show that the second order of convergence or close to that is achieved (in terms of sizes of finite volumes). Also it is shown that standard strategy, used by OpenFOAM is less efficient than the proposed approach. Finally it is concluded that for solving real problem a spatial adaptive mesh is essential and adaptive time steps also may be needed.

Numerical simulation of the conductivity relaxation in the high resistivity semiconductor

Abstract A theoretical model describing the relaxation of charge carriers in semiconductors of high resistance under the influence of the laser pulses is presented. It is demonstrated that parameters of the trapping states relevant to the processes of the conductivity relaxation can be defined by fitting the experimental data. Time evolution of the conductivity of the GaAs bulk semiconductor under the influence of nanosecond and picosecond laser pulses is considered. Effect of two laser pulses, when the first one results in population of the trapping state and the second one induces its depopulation, is also considered.

On Efficient Numerical Algorithms for Simulation of High Power Electrical Cables

AbstractThe new virtual modelling tool is constructed, which is used for optimal design of power transmission lines and cables. The construction of such lines should meet the latest power transmission network technical and economical requirements. The solver is is based on classical and modified mathematical models describing main heat conduction processes: diffusion, convection and radiation in various materials and environments. In basic heat conduction equation, we take into account a linear dependence of the resistance on temperature. Multi-physic and multi-scale models are required to simulate industrial cases of power transmission lines. The velocity of convective transport of the heat in air regions is simulated by solving a coupled thermo-convection problem including the heat conduction problem and the standard Navier-Stokes model of the heat flow in air. Another multi-physic model is used to describe changes of material heat conduction coefficients in soil due to influence of heating. This proces...

Numerical Simulation of Heat Transfer in Underground Electrical Cables

The aim of this project is to develop a virtual modelling tool which can be used to construct optimal design of power transmission lines and cables. They should meet the latest power transmission network technical and economical requirements. The mathematical model is based on a general heat conduction equation describing the diffusion, convection and radiation processes. We take into account a linear dependence of the resistance on temperature. The velocity of convective transport of the heat in air regions is obtained by solving a coupled thermoconvection problem including the heat conduction problem and a standard Navier-Stokes model of the flow in air. The changes of material coefficients in soil due to influence of heating are taken by solving a simplified mass balance equation for flows in porous media. The FVM is used to solve the obtained system of differential equations. Discretization of the domain is done by applying “aCute” mesh generator, which is a modification of the well-known Triangle mesh generator. The discrete schemes are implemented by using the OpenFOAM tool. Parallel versions of basic algorithms are also investigated. Results of computational experiments of simulation of real industrial underground cables are presented.

Numerical analysis of nonlinear model of excited carrier decay

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented.

Parallel algorithms for parabolic problems on graphs

In this paper two parallel numerical algorithms for solution of parabolic problems on graphs are investigated. The fully implicit and predictor-corrector finite difference schemes are proposed to approximate the differential equations on the given graph. The parallelization of the discrete algorithm is based on the domain decomposition method. Scalability analysis of the parallel algorithms is done. Some results of numerical simulations are presented and the efficiency of the proposed parallel algorithms is investigated.

Paralel Predictor-Corrector Schemes for Parabolic Problems on Graphs

Abstract We consider a predictor-corrector type finite difference scheme for solving one-dimensional parabolic problems. This algorithm decouples computations on different subdomains and thus can be efficiently implemented on parallel computers and used to solve problems on graph structures. The stability and convergence of the discrete solution is proved in the special energy and maximum norms. The results of computational experiments are presented.