A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

Axisymmetric Turbulent Methane Jet Propagation in a Co-Current Air Flow Under Combustion at a Finite Velocity

The paper introduces a numerical method for solving the problem of the axisymmetric methane jet propagation in an infinite co-current air flow. For modeling, we used the dimensionless equations of the turbulent boundary layer of reacting gases in the Mises coordinates. To close the Reynolds equation, a modified k - ε turbulence model was used. The k - ε model is considered a low Rhine turbulence model. Assuming that the intensities of convective and turbulent transfers of components are the same and using the stoichiometric ratios of the concentrations of components during combustion, we reduced five equations for the transfer and conservation of the mass of components to two equations for the relative excess concentration of the combustible gas. The concentrations of the components were determined from the solutions of these equations. By using relatively excessive velocities and total enthalpy, we reduced the boundary conditions for the three equations to a general form. To solve the problem in the Mises coordinates, we used a two-layer, six-point implicit finite-difference scheme, which provides the second order of accuracy of approximation in coordinates. The equations for the conservation and transfer of substances being non-linear, an iterative process was implemented. The influence of the radius of the fuel nozzle on the indices of the turbulent jet and flame was investigated. Findings of research show that in an endless co-current flow of fuel with a decrease in the radius of the nozzle, the rate of the chemical reaction and the maximum temperature in the calculation area decrease, and the amount of unburned part of the combustible gas increases

Block algorithms to solve Zheng/Chen/Zhang's finite-difference equations

This paper is devoted to the design of multiblock algorithms of the FDTD-method intended for computations based on a Zheng-Chen-Zhang implicit finite-difference scheme. Special emphasis is placed on experimental research of the designed algorithms and detecting specific features of the multiblock computing based on implicit finite-difference equations. The efficiency of the proposed approaches is proved by a six-fold speed-up of computations.

Numerical Simulation for a Fractal MIM Model for Solute Transport in Porous Media

A fractal mobile-immobile (MIM in short) model for solute transport in heterogeneous porous media is investigated from numerics. An implicit finite difference scheme is set forth for solving the coupled system, and stability and convergence of the scheme are proved based on the estimate of the spectral radius of the coefficient matrix. Numerical simulations with different parameters are presented to reveal the solute transport behaviors in the fractal case.

Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

Numerical Study Process Distribution of Contaminated Substances in the Atmosphere Taking Into Account the Physical and Mechanical Properties of Particles

The article deals with the numerical modeling of the processes of transfer and diffusion of air pollutants in the boundary layer of the atmosphere. A mathematical model of the spread of industrial emissions in the atmosphere was developed, taking into account the motion velocity of finely dispersed substances and a number of other factors affecting the change in the concentration of harmful substances in the atmosphere. The model is described by multidimensional partial differential equations with corresponding initial and boundary conditions. For the numerical integration of the problem, the method of splitting into physical processes (of transfer, diffusion and absorption) and an implicit finite-difference scheme of the second order of approximation in spatial variables and in time were used. A software tool was developed to conduct a computational experiment on a computer and to perform a comprehensive study of the processes of transfer and diffusion of harmful substances in the atmosphere