Explicit Difference Scheme Research Articles

Exact Finite Difference Schemes for Three-Dimensional Linear Systems with Constant Coefficients

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of $n$ linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.

On the convergence of an explicit difference scheme for evolution variational inequalities with nonlocal space operator

A nonlinear parabolic variational inequality with nonlocal space operator monotone with respect to gradient is considered. An explicit difference scheme with respect to the space operator and an implicit difference scheme with respect to the penalty operator are constructed by using the penalty method and the method of integral identities. Stability conditions for the constructed difference scheme are obtained. A convergence theorem with minimal assumptions on the smoothness of the initial data is proved.

Application of Computational Fluid Dynamics (CFD) in Contaminant Monitoring and Analysis

Computational fluid dynamics can be used to simulate the diffusion of pollutants and estimate the pollution load. So we can monitor and analysis the pollution better, then find the solution in time. In this paper, we combine the explicit difference scheme with the implicit difference scheme to solve the concentration convection-diffusion equation. The result shows that the absolute error between the numerical solutions and analytical solutions is very small. So the actual environmental problems can be simulated by Computational Fluid Dynamics numerical calculation. According to Computational Fluid Dynamics in the practical application of the environment, it has the characteristics of low cost, time saving and accurate. Therefore, the Computational Fluid Dynamics plays an important role in the pollutant analysis and monitoring.

The Order of Convergence of Difference Schemes for Fractional Equations

ABSTRACTIn this paper, we continue our research on convergence of difference schemes for fractional differential equations. Using implicit difference scheme and explicit difference scheme, we have a deal with the full discretization of the solutions of fractional differential equations in time variables and get the order of convergence.

Difference schemes in computations for chemically non-equilibrium processes in the rocket engine nozzles

The problem of determining the “potential” number of integration steps by explicit difference schemes is solved within the framework of the prediction model for chemically non-equilibrium processes.

The finite element method explicit scheme for a solution of one problem of surface and ground water combined movement

We investigate the initial-boundary value problem for the system of two nonlinear degenerate parabolic combined equations, when the one of equations is set in the domain we consider, and the second is set on the conducted section of this domain. With help of the semidiscretization methods with respect to time variable and finite element method (FEM) with respect to space variable we construct an explicit difference scheme. We obtained conditions which provide us convergence of constructed method. The result of convergence was obtained with the minimum conditions for smoothness of the initial data.

Numerical Simulation of a Second Order Traffic Flow Model

A second order fluid dynamical traffic flow model is considered which is a parabolic type second order partial differential equation appended with initial and boundary conditions reads as an initial boundary value problem (IBVP). For a linear velocity-density relation the paper presents the analytical solution of the model by using Cole-Hopf transformation. For the numerical solution of the IBVP, we investigate explicit central difference scheme and implement the numerical scheme by developing computer programming code. Error estimation is produced which shows the numerical solution is accurate up to eight decimal places and this result is much better than the work presented in [6]. The numerical feature of rate of convergence observed by graphical presentation is also found better than the previous work. The computed numerical simulation results are seen in a good qualitative agreement with the well known traffic flow behavior for various parameters.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 101-110

Boundary control problems for quasilinear systems of hyperbolic equations

For quasilinear systems of hyperbolic equations, the nonclassical boundary value problem of controlling solutions with the help of boundary conditions is considered. Previously, this problem was extensively studied in the case of the simplest hyperbolic equations, namely, the scalar wave equation and certain linear systems. The corresponding problem formulations and numerical solution algorithms are extended to nonlinear (quasilinear and conservative) systems of hyperbolic equations. Some numerical (grid-characteristic) methods are considered that were previously used to solve the above problems. They include explicit and implicit conservative difference schemes on compact stencils that are linearizations of Godunov’s method. The numerical algorithms and methods are tested as applied to well-known linear examples.

METHOD OF PARALLELIZATION OF NUMERICAL ALGORITHMOF NAVIER-STOKES EQUATIONS COMPLETE SYSTEM

The article considers the features of numerical construction of solutions of the Navier-Stokes equations full system describing a three-dimensional flow of compressible viscous heat-conducting gas under the action of gravity and Coriolis forces. It is shown that accounting of dissipative properties of viscosity and thermal conductivity of the moving continuum, even with constant coefficients of viscosity and thermal conductivity, as well as the use of explicit difference scheme calculation imposes significant restrictions on numerical experiments aimed at studying the arising complex flows of gas or liquid. First of all, it is associated with a signifi- cant complication of the system of equations, the restrictions on the value of the calculated steps in space and time, increasing the total computation time. One of the options is proposed of algorithm parallelization of numerical solution of the complete Navier - Stokes equations system in the vertical spatial coordinate. This parallelization option can significantly increase the computing performance and reduce the overall time of counting. A comparison of the results of calculation of one of options of gas flow in the upward swirling flow obtained by serial and parallel programs is presented.

Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural Hamiltonian systems: A comparative study of the accuracy of high-order schemes on molecular dynamics problems

The natural Hamiltonian systems (systems with separable Hamiltonians) are considered. The variety of explicit three-stage symplectic schemes is described. A classification of the third-order accurate schemes is given. All fourth-order schemes are found (there are seven of them). It is proved that there are no fifth-order schemes. The schemes with improved properties, such as invertibility and optimality with respect to the phase error, are listed. Numerical results that demonstrate the properties of these schemes are presented, and their comparative analysis with respect to the accuracy–efficiency criterion is given. The disbalance of total energy is used as the accuracy criterion.

PARALLEL COMPUTATIONS IN STUDIES OF DEPENDENCE OF GAS DYNAMIC PARAMETERS OF UPWARD SWIRLING FLOW OF GAS ON BLOWING VELOCITY

The rectangular parallelepiped explicit difference schemes for the numerical solution of the complete built system of Navier-Stokes equations. These solutions describe the three-dimensional flow of a compressible viscous heat-conducting gas in a rising swirling flows, provided the forces of gravity and Coriolis. This assumes constancy of the coefficient of viscosity and thermal conductivity. The initial conditions are the features that are the exact analytical solution of the complete Navier-Stokes equations. Propose specific boundary conditions under which the upward flow of gas is modeled by blowing through the square hole in the upper surface of the computational domain. A variant of parallelization algorithm for calculating gas dynamic and energy characteristics. The results of calculations of gasdynamic parameters dependency on the speed of the vertical blowing by the time the flow of a steady state flow.

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

AbstractAn explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.

An Efficient Method for Solving Spread Option Pricing Problem: Numerical Analysis and Computing

This paper deals with numerical analysis and computing of spread option pricing problem described by a two-spatial variables partial differential equation. Both European and American cases are treated. Taking advantage of a cross derivative removing technique, an explicit difference scheme is developed retaining the benefits of the one-dimensional finite difference method, preserving positivity, accuracy, and computational time efficiency. Numerical results illustrate the interest of the approach.

Моделирование неизотермической многофазной фильтрации с применением явных разностных схем

Моделирование неизотермической многофазной фильтрации с применением явных разностных схем

Analytic-Numerical Modeling and Investigation of Nanostructures' Dynamics on Material Surfaces After Laser Irradiation

<p class="R-AbstractKeywords">The work is devoted to modeling the formation and behavior of solid nano-sized particles on the surface of materials. In the simulation, it is assumed that the main processing technology of surface nanostructures is laser irradiation, which causes the Brownian motion of nanoparticles, due mainly to thermal fluctuations: if the temperature around the nanoparticles is uniformly distributed, the time average of the Brownian fluctuations is zero; however, if there is a temperature gradient around the nanoparticles, the thermal fluctuations affect the nanoparticle in different ways from different sides, and there is a force like the thermophoretic force, biasing the average position of the nanoparticle. When building a 1D model of the formation and flow behavior of nanoparticles, three important assumptions are introduced: the impact of nanoparticles on the process of irradiation is negligible; the impact of nanoparticles on each other as compared to the effect of laser irradiation on them is also negligible; and nanoparticles after laser irradiation can move both forward and backward and at every fixed period of time, moving the nanoparticles does not impose any steric constraints. Under the above assumptions, a 1D continuous model is built, implicit and explicit finite difference schemes to solve it are developed; their convergence and order of convergence are studied; an output condition ensuring the stability of the explicit difference scheme is obtained, the unconditional stability of implicit difference scheme is proven, and software for computer implementation of some of the obtained analytical and numerical results developed. </p>

X-FEM modeling of dynamic ductile fracture problems with a nonlocal damage-viscoplasticity model

In this paper, the dynamic large deformation X-FEM method is presented for modeling the full process of dynamic ductile fracture based on a nonlocal damage visco-plasticity model. The effect of inertia is modeled using an explicit central difference scheme which is enhanced through the use of mass lumping, reduced integration with hourglass control, and numerical damping. The material nonlinearity and the flow stress dependency on strain rate, hardening and temperature are modeled with the Johnson–Cook visco-plastic model. The micro-void nucleation, growth and coalescence are modeled macroscopically with an isotropic damage model. The localization phenomenon due to the damage and thermal softening is suppressed by using the visco-plastic regularization in combination with the nonlocal visco-plastic model. The large deformation and large strain formulations are implemented within the X-FEM framework to model the macro crack discontinuities using an updated Lagrangian approach. Crack propagation and crack direction criteria are presented and the issues relating to the combination of X-FEM with the damage visco-plasticity model are addressed. Finally, the robustness and accuracy of the proposed method are verified through several numerical examples.

Applying Explicit Schemes to the Korteweg-de Vries Equation

The water wave soliton is a result of a dynamic balance between dispersion and nonlinear effects. It brings together many branches of mathematics, some of which touch on deep ideas. The Korteweg-de Vries equation is typical of all model equations of nonlinear waves in the soliton phenomena. Four explicit difference schemes are used in order to approximate the Korteweg-de Vries equation, namely; (a) a First order upwind scheme, (b) the Zabusky-Kruskal scheme, (c) the Lax-Wendroff scheme, and (d) the Walkley scheme. Our main interest was to analyse which explicit scheme among the four performs well when implemented to the KdV equation to produce the best soliton solution. Hence, reviewing and considering existing schemes. Three sets of initial data are used to explore the numerical approximations. Two, including the data proposed by Zabusky and Kruskal, involve a single soliton wave, whilst the other involves the separation into two solitons, which will interact in time. These initial conditions and periodic boundary conditions are described in detail taking into account physical, mathematical and computational considerations. Accuracy, consistency and Fourier stability in regard to the four explicit schemes for the Korteweg-de Vries equation are discussed. Numerical results are reported for a single soliton solution and the separation into two solitons with different velocities are investigated. Graphical results are presented to show how well these four schemes agree well with each other. After comparing the four explicit schemes, the best scheme was the Zabusky and Kruskal scheme since it is a two-step scheme, which uses the explicit leapfrog finite difference scheme and was good for low amplitudes and less running time was needed than the other three explicit schemes.

Modeling of GPU computing using difference schemes

We propose models of GPU (graphics processing unit) computing that can be recommended for implementation of explicit and implicit difference schemes on GPUs. In particular, these are models for selecting an optimal vector length in the 'long-vector' algorithms and for finding an optimal pyramid height in corresponding parallel algorithms.

Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

Long vectors algorithms for solving grid equations of explicit difference schemes

We propose two variants of long vectors algorithms for solving grid equations of explicit difference schemes, allowing one to use the maximum number of CUDA cores even for a small dimension of the grid domain. The implementation of these algorithms is shown by the example of the heat conduction equation and the solution of Maxwell's equations. The comparison between the proposed algorithms implemented by means of the CUBLAS library and free software packages B-CALM and OpenCurrent is done.