Showing papers presented at "International Conference on Numerical Analysis and Its Applications in 2016"

Front Dynamics in an Activator-Inhibitor System of Equations

Abstract: We consider the construction of formal asymptotic approximation for solution of the singularly perturbed boundary value problem of an activator-inhibitor type with a solution in a form of moving front. Corresponding asymptotic analysis provides a priori information about the localization of the transition point for moving front that is further used for constructing of dynamic adapted mesh. This mesh significantly improves numerical stability of numerical calculations for the considered system.

Use of Asymptotics for New Dynamic Adapted Mesh Construction for Periodic Solutions with an Interior Layer of Reaction-Diffusion-Advection Equations

Abstract: This paper presents the development of analytic-numerical approaches to study periodically moving fronts in singularly perturbed reaction-diffusion-advection models. We describe the results of rigorous asymptotic treatment of the problem and suggest a method to generate a dynamic adapted mesh for the numerical solution of such problems. This method based on a priori information. In particular, we take into account a priori estimates on the location of the transition layer, its width and structure. An example is presented to demonstrate the effectiveness of the proposed method.

Combinatorial Modeling Approach to Find Rational Ways of Energy Development with Regard to Energy Security Requirements

Abstract: This paper describes the model, methods and tools to find rational ways of the energy development with regard to energy security requirements. As known, energy security is directly related to the uninterrupted energy supply. It is important to choose rational ways of the energy development with ensuring energy security in the future. A number of the specific external condition combinations of energy sector operation and development taking into account uncertainties and other factors leads to a huge possible energy sector states set. Therefore it cannot be processed in reasonable time. To overcome this issue an approach of combinatorial modeling is applied to manage the growing size of the energy sector states set.

A Generalized Multiscale Finite Element Method for Thermoelasticity Problems

Abstract: In this work, we consider the coupled systems of a partial differential equations, which arise in the modeling of thermoelasticity processes in heterogeneous domains. Heterogeneity of the properties requires a high resolution solve that adds many degrees of freedom that can be computationally costly. For the numerical solution, we use a Generalized Multiscale Finite Element Method (GMsFEM) that solves problem on a coarse grid by constructing local multiscale basis functions [1, 2, 3]. We construct multiscale basis functions for the temperature and for the displacements on the offline stage in each coarse block using local spectral problems [4, 5, 6, 7]. On the online stage we construct coarse scale system using precalculated multiscale basis functions and solve problem with any forcing and boundary conditions. The numerical results are presented for heterogeneous and perforated domains.

Regularization Methods of the Continuation Problem for the Parabolic Equation

Abstract: We investigate the one-dimensional continuation problem (the Cauchy problem) for the parabolic equation with the data on the part of the boundary. For numerical solution we apply finite-difference scheme inversion, the singular value decomposition and the gradient method of the minimizing the goal functional. The comparative analysis of numerical methods are presented.

Calculation of Kinetic Coefficients for Real Gases on Example of Nitrogen

Abstract: Problem of obtaining data on the properties of gaseous media is considered. Gases of interest are the gases used as transport systems in technical facilities. The focus is on calculating the kinetic coefficients of gaseous medium considering the molecular processes that take place in the gas flow. Molecular dynamics method is selected as the method of modeling. Various techniques for determining the kinetic coefficients of gases are described in detail and compared. The problem is considered on the example of nitrogen flow. For this goal calculating the coefficients of self-diffusion, shear viscosity and thermal conductivity for nitrogen is made. The obtained numerical results are in good agreement with known theoretical estimates and experimental data.

CVA Computing by PDE Models

Abstract: In order to incorporate the credit value adjustment (CVA) in derivative contracts, we propose a set of numerical methods to solve a nonlinear partial differential equation [2] modelling the CVA. Additionally to adequate boundary conditions proposals, characteristics methods, fixed point techniques and finite elements methods are designed and implemented. A numerical test illustrates the behavior of the model and methods.

Iterative Solution of the Retrospective Inverse Problem for a Parabolic Equation Using the Conjugate Gradient Method.

Abstract: In theory and practice of the inverse problems for unsteady partial differential equations, significant attention is paid to the problems of determination of the initial condition based on the values of the initial function in a finite time.

Existence and Stability of Contrast Structures in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems

Abstract: We consider stationary solutions with boundary and internal transition layers (contrast structures) for a nonlinear singularly perturbed equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an afficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions. The results can be used to create the numerical method which uses the asymptotic analyses to create space non uniform meshes to describe internal layer behavior of the solution.

Multi-preconditioned Domain Decomposition Methods in the Krylov Subspaces

Abstract: We consider the algebraic and geometric issues of the advanced parallel domain decomposition methods (DDMs) for solving very large non-symmetric systems of linear algebraic equations (SLAEs) that arise in the finite volume or the finite element approximation of the multi-dimensional boundary value problems on the non-structured grids. The main approaches in question for DDM include the balancing decomposition of the grid computational domain into parameterized overlapping or non-overlapping subdomains with different interface conditions on the internal boundaries. Also, we use two different sructures of the contacting the neigbour grid subdomains: with definition or without definition of the node dividers (separators) as the special grid subdomain. The proposed Schwarz parallel additive algorithms are based on the “total-flexible” multi-preconditioned semi-conjugate direction methods in the Krylov block subspaces. The acceleration of two-level iterative processes is provided by means of aggregation, or coarse grid correction, with different orders of basic functions, which realize a low - rank approximation of the original matrix. The auxiliary subsystems in subdomains are solved by direct or by the Krylov iterative methods. The parallel implementation of algorithms is based on hybrid programming with MPI-processes and multi-thread computing for the upper and the low levels of iterations, respectively. We describe some characteristic features of the computational technologies of DDMs that are realized within the framework of the library KRYLOV in the Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Novosibirsk. The technical requirements for this code are based on the absence of the program constraints on the degree of freedom and on the number of processor units. The conceptions of the creating the unified numerical envirement for DDMs are presented and discussed.

Inverse Problems of Determination of the Right-Hand Side Term in the Degenerate Higher-Order Parabolic Equation on a Plane

Abstract: We establish existence and uniqueness theorems as well as the theorem on stability under perturbations of the input data for the solution of the inverse problem for a degenerate higher-order parabolic equation on a plane with integral observation. We also obtain the estimates of the solution with constants explicitly written out in terms of the input data of the problem.

Computer Simulation of Plasma Dynamics in Open Plasma Trap

Abstract: We present a 2D particle-in-cell (PIC) model and the corresponding parallel code for computer simulation of plasma dynamics in open plasma traps. The mathematical model includes the Boltzmann equations for ions and electrons and system of Maxwell’s equations for the self-coordinate electromagnetic fields. The combination of the modified PIC-method and the Monte-Carlo methods is used to solve these equations. The problem of minimization of the plasma losses in trap with multipole magnetic walls has been investigated on the base of computer simulation.

Convergence Order of a Finite Volume Scheme for the Time-Fractional Diffusion Equation

Abstract: We consider the numerical approximation using the discrete gradient developed recently in the SUSHI method of [4] to approximate the time fractional diffusion equation in any space dimension. We derive and prove an error estimate in \(\mathbb {L}^\infty (\mathbb {L}^2)\)-norm.

Asymptotic-Numerical Method for the Location and Dynamics of Internal Layers in Singular Perturbed Parabolic Problems

Abstract: A singularly perturbed initial-boundary value problem for the parabolic reaction-diffusion-advection (RDA) equation is considered. Some effective asymptotic-numerical approach for the description of internal layers location and moving fronts dynamics is proposed.

Numerical Analysis of Reinforced Concrete Deep Beams

Abstract: In this work we consider numerical analysis of elasticity problem for reinforced concrete deep beams. Main investigation is made to define the effect of presence of steel-polypropylene fibres in concrete mixture for different types of reinforcement. For numerical solution we use finite element method approximation. Numerical realization of method performed on collection of free software FEniCS. As model problem we consider computation of elastically-deformed state of reinforced concrete structure, consisting of concrete matrix and steel reinforcement, loaded in 3-point bending test. Numerical results of three-dimensional problem with complex geometry are presented.

A Volunteer-Computing-Based Grid Architecture Incorporating Idle Resources of Computational Clusters

Abstract: In this paper, we suggest a new architecture of a computational grid that involves resources of BOINC-based volunteer computing projects and idle resources of computational clusters. We constructed a computational grid of the proposed kind, based on several computational clusters and the volunteer computing project SAT@home. This project, launched and maintained by us, is aimed at solving hard computational problems, which can be effectively reduced to Boolean satisfiability problem. In the constructed grid several new combinatorial designs based on diagonal Latin squares of order 10 were found, and also several weakened cryptanalysis problems for the Bivium cipher were solved.

On Some Piecewise Quadratic Spline Functions

Abstract: We study some \(C^1\) quadratic spline functions on bounded domain. The spline functions comprise polynomials, trigonometric functions, hyperbolic functions or their combinations. We show that some subset of minimal splines share most properties of the classical polynomial B-splines (positivity, compact support, smoothness, partition of unity). Some examples of polynomial and non-polynomial minimal splines are given.

Computation of Delta Greek for Non-linear Models in Mathematical Finance

Abstract: We consider a class of non-linear models in mathematical finance. The focus is on numerical study of Delta equation, where the unknown solution is the first spatial derivative of the option value. We also discuss the convergence to the viscosity solution. Newton’s and Picard’s iteration methods for solving the non-linear system of algebraic equations are proposed. Illustrative numerical examples are presented.

Hybrid Model of Particle Acceleration on a Shock Wave Front

Abstract: A new 2D hybrid numerical plasma model to investigate the processes of particle acceleration on a shock wave front is presented. This problem has a fundamental interest for astrophysics, plasma physics and charged particle accelerators. The model is based on the hybrid (or combined) approach where an electron component of plasma is described by the MHD-equations, while ions are treated kinetically via the Vlasov equation. One of the advantages of this approach is that it allows reduce the requirements for computing resources essentially comparing to a fully kinetic model. Another important advantage of it is the possibility to study the important instabilities on the ion time scale, neglecting high-frequency modes associated with electrons.

Discrete Approximations for Multidimensional Singular Integral Operators

Abstract: For discrete operator generated by singular kernel of Calderon–Zygmund one introduces a finite dimensional approximation which is a cyclic convolution. Using properties of a discrete Fourier transform and a finite discrete Fourier transform we prove a solvability for approximating equation in corresponding discrete space. For comparison discrete and finite discrete solution we obtain an estimate for a speed of convergence for a certain right-hand side of considered equation.

Conservative Finite-Difference Scheme for Computer Simulation of Field Optical Bistability

Abstract: We investigate 2D switching wave of nonlinear absorption in a semiconduntor under the high intensive laser pulse action. A laser pulse interaction with semiconductor is described by the set of 2D nonlinear differential equations. To solve these equations numerically we have developed the conservative finite-difference scheme. It’s realization is based on the original two-stage iteration process. It is very important, that the finite-difference scheme is conservative one on each of iterations because we have to provide a simulation on big time interval.

Numerical Modeling of Fluid Flow in Liver Lobule Using Double Porosity Model

Abstract: Earlier in the paper of Bonfiglio et al. [2] numerical simulation of blood circulation in the liver lobule was carried out using single porosity model. Electron microscopy reveals structure of the liver lobule, which has some of the properties of fractured porous media. In this work we consider double porosity model for modeling of blood filtration in the liver lobule. A numerical algorithm based on the spatial finite element approximation and finite difference approximation in time direction using explicit-implicit computational scheme is proposed.

A 2nd-Order FDM for a 2D Fractional Black-Scholes Equation

Abstract: We develop a finite difference method (FDM) for a 2D fractional Black-Scholes equation arising in the optimal control problem of pricing European options on two assets under two independent geometric Levy processes. We establish the convergence of the method by showing that the FDM is consistent, stable and monotone. We also show that the truncation error of the FDM is of 2nd order. Numerical experiments demonstrate that the method produces financially meaningful results when used for solving practical problems.

Numerical Method for Solving an Inverse Boundary Problem with Unknown Initial Conditions for Parabolic PDE Using Discrete Regularization

Abstract: We consider an inverse boundary values problem for parabolic PDE with unknown initial conditions. In this problem both Dirichlet and Neumann boundary conditions are given on a part of the boundary and it is required to determine the corresponding function on the remaining part of the boundary. To solve this problem, the numerical method based on finite difference schemes and regularization technique is proposed. The computing scheme involves solving the equation for each spatial step that allows to obtain the numerical solution in internal points of the domain and on the boundary. We prove a conditional stability of the method. The reliability and the efficiency of the method were confirmed by computational results.

Numerical Methods for a Class of Fractional Advection-Diffusion Models with Functional Delay

Abstract: In this paper, we consider a technique of creation of difference schemes for time and space fractional partial differential equations with effect of delay on time. For two sided space fractional diffusion equation and fractional advection equations with time functional after-effect, an implicit numerical method is constructed. We use shifted Grunwald-Letnikov formulae to approximate space fractional derivatives and L1-algorithm to approximate time fractional derivatives. We also use piecewise constant interpolation and extrapolation by continuation for the prehistory of model with respect to time. The algorithm is a fractional analogue of the pure implicit numerical method in which the model is reduced on each time step to the solution of linear algebraic system. The order of convergence is obtained. Numerical experiments are carried out to support the obtained theoretical results.

Numerical Simulation of Heat Transfer of the Pile Foundations with Permafrost

Abstract: In this work we consider the numerical simulation of the dynamics of soil temperature in a permafrost area. The mathematical formulation of the problem with appropriate initial and boundary conditions is presented. A computational algorithm is based on the finite element approximation in space. To approximate in time we use the standard fully implicit scheme with linearisation from previous time layer. We present results of research of temperature stabilization time and the impact of presence of piles on the temperature of the surrounding soil. Numerical comparisons of two-dimensional and three-dimensional model problems are presented.

Simulation of Surface Heating Process with Laser

Abstract: The model of heating of surface of solid substance with infra-red laser and cooling with air flow is created. From initial differential partial equations the discrete form of equation is obtained. The method of integration of system under dynamic change of energy of laser beam and constant parameters of airflow is worked out. Due to this method the initial model was transformed into a discrete form. The discrete equation is solved for the case of dynamically changing laser beam energy and constant parameters of the airflow. The result of solving of equation is graphed in 3D space: distance from center of laser beam/distance from surface of target/temperature.

A Singularly Perturbed Boundary Value Problems with Fractional Powers of Elliptic Operators

Abstract: A boundary value problem for a fractional power \(0< \varepsilon < 1\) of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when \(\varepsilon \rightarrow 0\). It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimensional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with \(0 < \varepsilon \ll 1\).

The Service-Oriented Multiagent Approach to High-Performance Scientific Computing

Abstract: The tools for intelligent management of high-performance computing in a heterogeneous distributed computing environment for solving large scientific problems are represented and the service-oriented multiagent approach to solve such problems using these tools is proposed. A purpose of our research is expansion of opportunities for management of the considered environment. Advantages of the proposed approach as compared with approaches based on use of the traditional systems for a distributed computing management are illustrated with two examples of scientific services. Experimental results show a high scalability and efficiency for calculations carried out with use of these services.

Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation

Abstract: In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order \(\delta \) where \(0<\delta <1\). It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time \(t=0\); this reduces the global order of convergence of the finite difference method to \(O(h^2+\tau ^\delta )\), where h and \(\tau \) are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from \(t=0\). This motivates us to investigate if the finite difference method is more accurate away from \(t=0\). Numerical experiments with various non-smooth and incompatible initial conditions show that, away from \(t=0\), one obtains \(O(h^2+\tau )\) convergence.