Showing papers presented at "International Conference on Numerical Analysis and Its Applications in 2012"

Comparison Principle for Reaction-Diffusion-Advection Problems with Boundary and Internal Layers

Abstract: In the present paper we discuss father development of the general scheme of the asymptotic method of differential inequalities and illustrate it applying for some new important cases of initial boundary value problem for the nonlinear singularly perturbed parabolic equations,which are called in applications as reaction-diffusion-advection equations. The theorems which state front motion description and stationary contrast structures formation are proved for parabolic, parabolic-periodic and integro-parabolic problems.

Asynchronous Differential Evolution with Restart

Abstract: Asynchronous Differential Evolution ADE [1] is a derivative-free method to solve global optimization problems. It provides effective parallel realization. In this work we derive ADE with restart ADE-R. By increasing population size after each restart, new strategy enhances its chances to locate the global minimum. The ADE-R algorithm has convergence rate comparable or better than ADE with fixed population sizes. Performance of the ADE-R algorithm is demonstrated on a set of benchmark functions.

Asymptotic-numerical Investigation of Generation and Motion of Fronts in Phase Transition Models

Abstract: We propose an effective asymptotic-numerical approach to the problem of moving front type solutions in nonlinear reaction-diffusion-advection equations. The dimension of spatial variables for the location of a moving front is lower per unit then the original problem. This fact gives the possibility to save computing resources in numerical experiments and speed up the process of constructing approximate solutions with a suitable accuracy.

Variance-Based Sensitivity Analysis of the Unified Danish Eulerian Model According to Variations of Chemical Rates

Abstract: A special computational technology for sensitivity analysis of ozone concentrations according to variations of rates of chemical reactions is developed. It allows us to study a larger number of reactions than we have considered in our previous study. The reactions are taken from the standardized scheme for air-pollution chemistry CBM-IV. A number of numerical experiments with a large-scale air pollution model Unified Danish Eulerian Model, UNI-DEM have been carried out to compute Sobol sensitivity measures. The sensitivity study has been done for the areas of four European cities Genova, Milan, Manchester, and Edinburgh with different geographical locations.

Interior Layers in Coupled System of Two Singularly Perturbed Reaction-Diffusion Equations with Discontinuous Source Term

Abstract: We study a coupled system of two singularly perturbed linear reaction-diffusion equations with discontinuous source term. A central difference scheme on layer-adapted piecewise-uniform mesh is used to solve the system numerically. The scheme is proved to be almost first order uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical results are presented to support the theoretical results.

Symplectic Numerical Schemes for Stochastic Systems Preserving Hamiltonian Functions

Abstract: We present high-order symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions. The approach is based on the generating function method, and we show that for the stochastic Hamiltonian systems, the coefficients of the generating function are invariant under permutations. As a consequence, the high-order symplectic schemes have a simpler form than the explicit Taylor expansion schemes with the same order. Moreover, we demonstrate numerically that the symplectic schemes are effective for long time simulations.

Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H -s , 0 ≤ s ≤ 1

Abstract: We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data We assume that Ωi¾?i¾?i¾?ℝ d , di¾?=i¾?1,2,3 is a convex polygonal polyhedral domain We theoretically justify optimal order error estimates in L 2- and H 1-norms for initial data in H -i¾?s Ω,i¾?0i¾?≤i¾?si¾?≤i¾?1 We confirm our theoretical findings with a number of numerical tests that include initial data v being a Dirac i¾?-function supported on a di¾?-i¾?1-dimensional manifold

Boundary Value Problems for Fractional PDE and Their Numerical Approximation

Abstract: Fractional order partial differential equations are considered. The main attention is devoted to fractional in time diffusion equation. An interface problem for this equation is studied and its well posedness in the corresponding Sobolev like spaces is proved. Analogous results are obtained for a transmission problem in disjoint intervals.

Runge-Kutta Methods with Equation Dependent Coefficients

Abstract: The simplest two and three stage explicit Runge-Kutta methods are examined by a conveniently adapted form of the exponential fitting approach. The unusual feature is that the coefficients of the new versions are no longer constant, as in standard versions, but depend on the equation to be solved. Some valuable properties emerge from this. Thus, in the case of three-stage versions, although in general the order is three, that is the same as for the standard method, this is easily increased to four by a suitable choice of the position of the stage abscissas. Also, the stability properties are massively enhanced. Two particular versions of order four are A-stable, a fact which is quite unusual for explicit methods. This recommends them as efficient tools for solving stiff differential equations.

Positivity Preserving Numerical Method for Non-linear Black-Scholes Models

Abstract: A motivation for studying the nonlinear Black- Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile unprotected portfolio. In this work we develop positivity preserving algorithm for solving a large class of non-linear models in mathematical finance on the original infinite domain. Numerical examples are discussed.

Numerical Study of Maximum Norm a Posteriori Error Estimates for Singularly Perturbed Parabolic Problems

Abstract: A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered. Certain critical details of the implementation are addressed. Based on numerical results we discuss various aspects of the error estimators in particular their effectiveness.

Comparison of Two Numerical Approaches to Boussinesq Paradigm Equation

Abstract: In order to study the time behavior and structural stability of the solutions of Boussinesq Paradigm Equation, two different numerical approaches are designed. The first one A1 is based on splitting the fourth order equation to a system of a hyperbolic and an elliptic equation. The corresponding implicit difference scheme is solved with an iterative solver. The second approach A2 consists in devising of a finite difference factorization scheme. This scheme is split into a sequence of three simpler ones that lead to five-diagonal systems of linear algebraic equations. The schemes, corresponding to both approaches A1 and A2, have second order truncation error in space and time. The results obtained by both approaches are in good agreement with each other.

High Order Accurate Difference Schemes for Hyperbolic IBVP

Abstract: In the present paper the initial-boundary value problem for multidimensional hyperbolic equation with Dirichlet condition is considered. The third and fourth orders of accuracy difference schemes for the approximate solution of this problem are presented and the stability estimates for the solutions of these difference schemes are obtained. Some results of numerical experiments are presented in order to support theoretical statements.

Numerical Methods for Evolutionary Equations with Delay and Software Package PDDE

Abstract: The paper gives a survey of the author's results on the grid-based numerical algorithms for solving the evolutionary equations parabolic and hyperbolic with the effect of heredity on a time variable. From uniform positions we construct analogs of schemes with weights for the one-dimensional heat conduction equation with delay of general form, analog of a method of variable directions for the equation of parabolic type with time delay and two spatial variables, analog of the scheme with weights for the equation of hyperbolic type with delay. For the one-dimensional heat conduction equation and the wave equation we obtained conditions on the weight coefficients that ensure stability on the prehistory of the initial function. Numerical algorithms are implemented in the form of software package Partial Delay Differential Equations PDDE toolbox.

Method SMIF for Incompressible Fluid Flows Modeling

Abstract: For solving of the Navier-Stokes equations describing 3D incompressible viscous fluid flows the Splitting on physical factors Method for Incompressible Fluid flows SMIF with hybrid explicit finite difference scheme second-order accuracy in space, minimum scheme viscosity and dispersion, capable for work in the wide range of Reynolds Re and internal Froude Fr numbers and monotonous based on the Modified Central Difference Scheme and the Modified Upwind Difference Scheme with a special switch condition depending on the velocity sign and the signs of the first and second differences of the transferred functions has been developed and successfully applied. At the present paper the description of the numerical method SMIF and it's application for simulation of the 3D separated homogeneous and density stratified fluid flows around a sphere are demonstrated.

The Finite Element Method for Boundary Value Problems with Strong Singularity and Double Singularity

Abstract: A boundary value problem is said to possess strong singularity if its solution u does not belong to the Sobolev space $W^1_2$ H 1 or, in other words, the Dirichlet integral of the solution u diverges. We consider the boundary value problems with strong singularity and with double singularity caused the discontinuity of coefficients in the equation on the domain with slot and presence of the corners equal 2π on boundary of this domain. The schemes of the finite element method is constructed on the basis of the definition on R i¾? -generalized solution to these problems, and the finite element space contains singular power functions. The rate of convergence of the approximate solution to the R i¾? -generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.

Uniform Shift Estimates for Transmission Problems and Optimal Rates of Convergence for the Parametric Finite Element Method

Abstract: Let Ωi¾?i¾?i¾?ℝ d , $d \geqslant 1$ , be a bounded domain with piecewise smooth boundary i¾?i¾?Ω and let U be an open subset of a Banach space Y. Motivated by questions in "Uncertainty Quantification," we consider a parametric family Pi¾?=i¾?P y yi¾?∈i¾?U of uniformly strongly elliptic, second order partial differential operators P y on Ω. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution u: Ω×Ui¾?i¾?i¾?ℝ of the parametric, elliptic boundary value/transmission problem P y u y i¾?=i¾?f y , yi¾?∈i¾?U, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for di¾?=i¾?2. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces [InlineEquation not available: see fulltext.] of Babuska-Kondrat'ev type in Ω, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs P y yi¾?∈i¾?U admit a shift theorem that is uniform in the parameter yi¾?∈i¾?U. In turn, this then leads to h m -quasi-optimal rates of convergence i. e., algebraic orders of convergence for the Galerkin approximations of the solution u, where the approximation spaces are defined using the "polynomial chaos expansion" of u with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab 2010.

A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

Abstract: A singularly perturbed reaction-diffusion parabolic problem with an incompatibility between the initial and boundary conditions is examined. A finite difference scheme is considered which utilizes a special finite difference operator and a piecewise uniform Shishkin mesh. Numerical results are presented for both nodal and global pointwise convergence, using bilinear interpolation and, also, an interpolation method based on the error function. These results show that the method is not globally convergent when bilinear interpolation is used but they indicate that, for the test problem considered, it is globally convergent using the second type of interpolation.

Stability of Implicit Difference Scheme for Solving the Identification Problem of a Parabolic Equation

Abstract: We consider the inverse problem of reconstructing the right side of a parabolic equation with an unknown time dependent source function. Numerical solution and well-posedness of this type problem with local boundary conditions considered previously by A.A. Samarskii, P.N. Vabishchevich and V.T. Borukhov. In this paper, we focus on studying the stability of the problem with nonlocal conditions. A stable algorithm for the approximate solution of the problem is presented.

Quadrature Formula with Five Nodes for Functions with a Boundary Layer Component

Abstract: Quadrature formula for one variable functions with a boundary layer component is constructed and studied It is assumed that the integrand can be represented as a sum of regular and boundary layer components The boundary layer component has high gradients, therefore an application of Newton-Cotes quadrature formulas leads to large errors An analogue of Newton-Cotes rule with five nodes is constructed The error of the constructed formula does not depend on gradients of the boundary layer component Results of numerical experiments are presented

Finite Volume Approximations for Incompressible Miscible Displacement Problems in Porous Media with Modified Method of Characteristics

Abstract: The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. The pressure-velocity is elliptic type and the concentration equations is convection dominated diffusion type. It is known that miscible displacement problems follow the natural law of conservation and finite volume methods are conservative. Hence, in this paper, we present a mixed finite volume element method FVEM for the approximation of the pressure-velocity equation. Since concentration equation is convection dominated diffusion type and most of the numerical methods suffer from the grid orientation effect and modified method of characteristicsMMOC minimizes the grid orientation effect. Therefore, for the approximation of the concentration equation we apply a standard FVEM combined MMOC. A priori error estimates are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.

On a Research of Hybrid Methods

Abstract: Constructed hybrid methods of the high accuracy the experts examined that's for solving integral and integro-differential equations. Using hybrid methods for solving integral equations belongs to Makroglou. Here, developing these idea, explored a more general hybrid method which is applied to solving Volterra integral equations and also constructed a concrete method with the degree pi¾?=i¾?8. However, order of accuracy for the known corresponding methods is of level pi¾?≤i¾?4.

Note on the Convergence of the Implicit Euler Method

Abstract: For the solution of the Cauchy problem for the first order ODE, the most popular, simplest and widely used method are the Euler methods. The two basic variants of the Euler methods are the explicit Euler methods EEM and the implicit Euler method IEM. These methods are well-known and they are introduced almost in any arbitrary textbook of the numerical analysis, and their consistency is given. However, in the investigation of these methods there is a difference in concerning the convergence: for the EEM it is done almost everywhere but for the IEM usually it is missed. E.g., [1, 2, 6-9].The stability and hence, the convergence property of the IEM is usually shown as a consequence of some more general theory. Typically, from the theory for the implicit Runge-Kutta methods, which requires knowledge of several basic notions in numerical analysis of ODE theory, and the proofs are rather complicated. In this communication we will present an easy and elementary prove for the convergence of the IEM for the scalar ODE problem. This proof is direct and it is available for the non-specialists, too.

On a Mathematical Model of Adaptive Immune Response to Viral Infection

Abstract: In this paper we study a mathematical model formulated within the framework of the kinetic theory for active particles The model is a bilinear system of integro-differential equations IDE of Boltzmann type and it describes the interactions between virus population and the adaptive immune system The population of cytotoxic T lymphocytes is additionally divided into precursor and effector cells Conditions for existence and uniqueness of the solution are studied Numerical simulations of the model are presented and discussed

Finite Difference Scheme for a Parabolic Transmission Problem in Disjoint Domains

Abstract: In this paper we investigate a parabolic transmission problem in disjoint domains. A priori estimate for its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.

A Multicomponent Alternating Direction Method for Numerical Solution of Boussinesq Paradigm Equation

Abstract: We construct and analyze a multicomponent alternating direction method a vector additive scheme for the numerical solution of the multidimensional Boussinesq Paradigm Equation BPE In contrast to the standard splitting methods at every time level a system of many finite difference schemes is solved Thus, a vector of the discrete solutions to these schemes is found It is proved that these discrete solutions converge to the continuous solution in the uniform mesh norm with O|h|2i¾?+i¾?i¾? order The method provides full approximation to BPE and is efficient in implementation The numerical rate of convergence and the altitudes of the crests of the traveling waves are evaluated

A Fourth-Order Iterative Solver for the Singular Poisson Equation

Abstract: A compact fourth-order finite difference scheme solver devoted to the singular-Poisson equation is proposed and verified The solver is based on a mixed formulation: the Poisson equation is splitted into a system of partial differential equations of the first order This system is then discretized using a fourth-order compact scheme This leads to a sparse linear system but introduces new variables related to the gradient of an unknow function The Schur factorization allows us to work on a linear sub-problem for which a conjugated-gradient preconditioned by an algebraic multigrid method is proposedNumerical results show that the new proposed Poisson solver is efficient while retaining the fourth-order compact accuracy

Finite Element Simulation of Nanoindentation Process

Abstract: Recently, nanoindentation technique is gaining importance in determination of the mechanical parameters of thin films and coatings. Most commonly, the instrumented indentation data are used to obtain two material characteristics of bulk materials: indentation modulus and indentation hardness. In this paper the authors discuss the possibility by means of numerical simulations of nanoindentation tests to obtained the force-displacement curve employing various constitutive models for both the substrate and the coating. Examples are given to demonstrate the influence of some features of the numerical model and the model assumptions on the quality of the simulation results. The main steps in creation of the numerical model and performing the numerical simulation of nanoindentation testing process are systematically studied and explained and the conclusions are drawn.

On the Unsymmetrical Buckling of the Nonuniform Orthotropic Circular Plates

Abstract: This work is concerned with the numerical study of unsymmetrical buckling of clamped orthotropic plates under uniform pressure. The effect of material heterogeneity on the buckling load is examined. The refined 2D shell theory is employed to obtain the governing equations for buckling of a clamped circular shell. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. It is shown that if the elasticity modulus decreases away from the center of a plate, the critical pressure for unsymmetric buckling is sufficiently lower than for a plate with constant mechanical properties.

The Numerical Solution of the Boundary Function Inverse Problem for the Tidal Models

Abstract: The problem of propagation of long waves in a domain of an arbitrary form with the sufficiently smooth boundary on a sphere is considered. The boundary consists of "solid" parts passing along the coastline and "open liquid" parts passing through the water area. In general case the influence of the ocean through an open boundary is unknown and must be found together with components of a velocity vector and free surface elevation. For this purpose we use observation data of free surface elevation given only on a part of an "open liquid" boundary. We solve our ill-posed inverse problem by an approach based on the optimal control methods and adjoint equations theory.