149 research outputs found
Didelės skaičiavimo apimties neklasikinių uždavinių sprendimo lygiagretieji algoritmai.
No full textThis research focuses on parallel algorithms, which help to solve limited memory and computational time problems. In the second chapter of the dissertation parallel algorithms for various most recent numerical methods were studied and compared. These methods are based on the general approach: the given non-local differential problem with fractional powers of the Laplacian is transformed to a local differential problem of elliptic or pseudo-parabolic type, but formulated in a higher dimensional space R^(d+1), if Ω ⊂ R^d . The scalability and convergence analysis of parallel algorithms was performed. Recommendations to achieve a given accuracy for the provided fractional power coefficient were specified. In the third chapter the detailed analysis of absorbing boundary conditions for the linear Schrödinger equation was performed. Recommendations for constructing absorbing boundary conditions for the one-dimensional Schrödinger equation using methods based on the approximation of exact transparent boundary conditions by rational functions were presented. The proposed methodology has shown, that it is possible to find the accurate absorbing boundary conditions for four qualitatively different tasks. In the fourth chapter a three-level parallelisation scheme was proposed. The possibilities of this methodology are demonstrated for solving local optimization problems. The proposed three-level scheme increases the amount of computational resources, which can be used efficiently
NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION EQUATION
No full textHyperbolic heat conduction problem is solved numerically. The explicit and implicit Euler schemes are constructed and investigated. It is shown that the implicit Euler scheme can be used to solve efficiently parabolic and hyperbolic heat conduction problems. This scheme is unconditionally stable for both problems. For many integration methods strong numerical oscillations are present, when the initial and boundary conditions are discontinuous for the hyperbolic problem. In order to regularize the implicit Euler scheme, a simple linear relation between time and space steps is proposed, which automatically introduces sufficient amount of numerical viscosity. Results of numerical experiments are presented.
First published online: 14 Oct 201
Parallel algorithms for non-classical problems with big computational costs.
No full textThis research focuses on parallel algorithms, which help to solve limited memory and computational time problems. In the second chapter of the dissertation parallel algorithms for various most recent numerical methods were studied and compared. These methods are based on the general approach: the given non-local differential problem with fractional powers of the Laplacian is transformed to a local differential problem of elliptic or pseudo-parabolic type, but formulated in a higher dimensional space R^(d+1), if Ω ⊂ R^d . The scalability and convergence analysis of parallel algorithms was performed. Recommendations to achieve a given accuracy for the provided fractional power coefficient were specified. In the third chapter the detailed analysis of absorbing boundary conditions for the linear Schrödinger equation was performed. Recommendations for constructing absorbing boundary conditions for the one-dimensional Schrödinger equation using methods based on the approximation of exact transparent boundary conditions by rational functions were presented. The proposed methodology has shown, that it is possible to find the accurate absorbing boundary conditions for four qualitatively different tasks. In the fourth chapter a three-level parallelisation scheme was proposed. The possibilities of this methodology are demonstrated for solving local optimization problems. The proposed three-level scheme increases the amount of computational resources, which can be used efficiently
Baigtinių tūrių kintamųjų krypčių schemos hibridinės dimensijos šilumos laidumo modeliams.
No full textIn this dissertation, the method of dimension reduction is applied to some mathematical models. Due to it, the time required to find a solution to a differential equation is reduced a few times, however, some accuracy is lost this way. Two models of heat conduction are presented, where in some part of domain the dimension is reduced from 3 or 2 to 1. Then a FVM ADI numerical scheme is justified for obtained hybrid dimension models. Some important properties of this scheme are proved. Due to nonstandard additional conditions, the classical methods are modified. A unique existence of a numerical solution is proved. Also, a fourth order partial differential equation is investigated, which was derived from a model of viscous fluid flowing through an elastic vessel. Here the dimension is reduced to 1 in the whole domain. The specification of problem is discussed along with some numerical schemes and their properties. For the problems solved in this dissertation, provided numerical experiments agree well with theoretical estimates and justify the practical usage of constructed schemes
Numerical algorithms for one parabolic-elliptic problem.
No full textIn this paper we solve numerically a parabolic-elliptic problem.Two finite difference schemes are proposed. The first scheme is a modification of the backward Euler algorithm and it requires to solve an elliptic problem at each time step.The spectral estimates of the obtained matrix are presented. The second scheme is a modification of the stability-correction scheme. This scheme is used as a classical splitting scheme in the parabolic region of the problem definition and as a new iterative algorithm in the elliptic part of the problem. We prove the convergence of the proposed scheme
Numerical algorithms for solving the optimal control problem of simple bioreactors
No full textThe modified nonlocal feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion–convection–reaction equations. The analysis of the influence of the convection transport is one the aims of this paper. The control loop is defined using the relation, which shows how the amount of the drug produced in the bioreactor and delivered into a human body depends on the substrate concentration specified on the external boundary of the bioreactor. The system of PDEs is solved by using the finite volume and finite difference methods, the control loop parameters are defined from the analysis of stationary linearized equations. The second aim of this paper is to solve the inverse problem and to determine optimal boundary conditions. These results enable us to estimate the potential accuracy of the proposed devices.
 
On the aposteriori error estimates for finite‐element solutions
No full textThe analysis of the accuracy of the aposteriori error estimation procedure for finite‐element solutions is presented. The function Y — y is used as an aposteriori error estimator, here y ∈ S 0 1,Δ is the finite‐element solution of the given problem and Y ∈ S 0 2,Δis the high order solution of the same problem. The second order accuracy is proved for this error estimator in the L 2, H 1 and L8 norms. Results of numerical experiments are presented.
First Published Online: 14 Oct 201
Editor\u27s letter
No full text„Editor\u27s letter" Mathematical Modelling and Analysis, 14(1), p. 13
Editor\u27s letter
No full text„Editor\u27s letter" Mathematical Modelling and Analysis, 14(1), p. 13
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