The Investigation of Efficiency of Physical Phenomena Modelling Using Differential Equations on Distributed Systems

This work is dedicated to development of mathematical modelling software. In this dissertation numerical methods and algorithms are investigated in software making context. While applying a numerical method it is important to take into account the limited computer resources, the architecture of these resources and how do methods affect software robustness. Three main aspects of this investigation are that software implementation must be efficient, robust and be able to utilize specific hardware resources. The hardware specificity in this work is related to distributed computations of different types: single CPU with multiple cores, multiple CPUs with multiple cores and highly parallel multithreaded GPU device. The investigation is done in three directions: GPU usage for 3D FDTD calculations, FVM method usage to implement efficient calculations of a very specific heat transferring problem, and development of special techniques for software for specific bacteria self organization problem when the results are sensitive to numerical methods, initial data and even computer round-off errors. All these directions are dedicated to create correct technological components that make a software implementation robust and efficient. The time prediction model for 3D FDTD calculations is proposed, which lets to evaluate the efficiency of different GPUs. A reasonable speedup with GPU comparing to CPU is obtained. For FVM implementation the OpenFOAM open source software is selected as a basis for implementation of calculations and a few algorithms and their modifications to solve efficiency issues are proposed. The FVM parallel solver is implemented and analyzed, it is adapted to heterogeneous cluster Vilkas. To create robust software for simulation of bacteria self organization mathematically robust methods are applied and results are analyzed, the algorithm is modified for parallel computations

Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers

The grid approximation of an initial‐boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation. The second‐order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters å 2 1 and å 2 2, respectively, that take arbitrary values in the open‐closed interval (0,1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial‐boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor‐product grid, piecewise‐uniform in xand t, a difference scheme is constructed that converges å‐uniformly at the rate O(N−2 ln2 N + N0 −1 ln N0 ), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively. First Published Online: 14 Oct 201

An a-posteriori adaptive mesh technique for singularly perturbed convection-diffusion problems with a moving interior layer

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not~converge $eps$-uniformly in the uniform norm, even under the condition $N^{-1}+N_0^{-1}approx eps$, where $eps$ is the perturbation parameter and $N$ and $N_0$ denote the number of mesh points with respect to $x$ and $t$. In the case of rectangular meshes which are ({it a~priori,} or {it a~posteriori,}) locally refined in the transition layer, there are no schemes that convergence uniformly in $eps$ even under the {it very restrictive,} condition $N^{-2}+N_0^{-2} approx eps$. However, the condition for convergence can be {it essentially weakened} if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an {it a~priori,}, or an {it a~posteriori,} adaptive mesh technique. Here we construct a scheme on {it a~posteriori,} adaptive meshes (based on the gradient of the solution), whose solution converges `almost $eps$-uniformly', viz., under the condition $N^{-1}=o(eps^{ u})$, where $u>0$ is an arbitrary number from the half-open interval $(0,1]$

Fitted numerical methods to solve di erential models describing unsteady magneto-hydrodynamic ow

Philosophiae Doctor - PhDIn this thesis, we consider some nonlinear di erential models that govern unsteady magneto-hydrodynamic convective ow and mass transfer of viscous, incompressible, electrically conducting uid past a porous plate with/without heat sources. The study focusses on the e ect of a combination of a number of physical parameters (e.g., chem- ical reaction, suction, radiation, soret e ect, thermophoresis and radiation absorption) which play vital role in these models. Non-dimensionalization of these models gives us sets of di erential equations. Reliable solutions of such di erential equations can- not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a tted operator nite di erence method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame-ters, we present extensive numerical simulations for each of these models. Finally, we con rm theoretical results through a set of speci c numerical experiments

Fitted numerical methods to solve differential models describing unsteady magneto-hydrodynamic flow

Philosophiae Doctor - PhDIn this thesis, we consider some nonlinear differential models that govern unsteady magneto-hydrodynamic convective flow and mass transfer of viscous, incompressible,electrically conducting fluid past a porous plate with/without heat sources. The study focusses on the effect of a combination of a number of physical parameters (e.g., chemical reaction, suction, radiation, soret effect,thermophoresis and radiation absorption) which play vital role in these models.Non dimensionalization of these models gives us sets of differential equations. Reliable solutions of such differential equations can-not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a fitted operator finite difference method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame- ters, we present extensive numerical simulations for each of these models. Finally, we confirm theoretical results through a set of specificc numerical experiments

A Multiscale Thermo-Fluid Computational Model for a Two-Phase Cooling System

In this paper, we describe a mathematical model and a numerical simulation method for the condenser component of a novel two-phase thermosyphon cooling system for power electronics applications. The condenser consists of a set of roll-bonded vertically mounted fins among which air flows by either natural or forced convection. In order to deepen the understanding of the mechanisms that determine the performance of the condenser and to facilitate the further optimization of its industrial design, a multiscale approach is developed to reduce as much as possible the complexity of the simulation code while maintaining reasonable predictive accuracy. To this end, heat diffusion in the fins and its convective transport in air are modeled as 2D processes while the flow of the two-phase coolant within the fins is modeled as a 1D network of pipes. For the numerical solution of the resulting equations, a Dual Mixed-Finite Volume scheme with Exponential Fitting stabilization is used for 2D heat diffusion and convection while a Primal Mixed Finite Element discretization method with upwind stabilization is used for the 1D coolant flow. The mathematical model and the numerical method are validated through extensive simulations of realistic device structures which prove to be in excellent agreement with available experimental data