Order Compact Finite Difference Method Research Articles

High order compact split step finite difference method for two-dimensional coupled nonlinear Schrödinger system

In this paper, a decoupled and efficient finite difference method is developed for two-dimensional coupled nonlinear Schrödinger (CNLS) system. The proposed method uses split step technique for the temporal discretization and high order compact (HOC) difference approximation for the spatial discretization. The original problem is decomposed into a two-dimensional linear subproblem and a two-dimensional nonlinear subproblem. For the two-dimensional linear subproblem, the Lie-Trotter splitting formula is adopted in time to reduce computational cost. While for the nonlinear subproblem, it can be integrated directly and exactly. By the von Neumann approach, it is showed that the proposed method is unconditionally stable. Numerical examples are conducted to compare it with other scheme and numerical results verified the superiority of the proposed method in terms of accuracy and efficiency. The new method also exhibits good numerical performance in long-time simulation.

The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

<abstract><p>The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.</p></abstract>

Numerical Solution of Two-Parameter Singularly Perturbed Convection-Diffusion Boundary Value Problems via Fourth Order Compact Finite Difference Method

In this study, we have developed a fourth order compact finite difference method for finding the numerical solution of two-parameter singularly perturbed convection-diffusion boundary value problems. We have used fourth order compact finite difference method on uniform mesh which provides a tridiagonal linear system of equations. The convergence analysis of the proposed method is established through a matrix analysis approach and it is proved that present method gives fourth order convergence results. Present method is implemented on two numerical examples for checking the efficiency and precision of the method. Numerical outcomes are exhibited which supports the theoretical outcomes. Numerical outcomes are compared with other existing methods and found that present method gives more accurate approximate solution as compare to the other existing methods.

A new higher order compact finite difference method for generalised Black–Scholes partial differential equation: European call option

This paper presents a high order numerical method based on a uniform mesh to obtain a highly accurate result for generalized Black–Scholes equation arising in the financial market. We first semi-discretize the underlying problem in the temporal direction by using Crank–Nicolson scheme and then discretize the resulting set of equations by using a high order compact finite difference method (HOCFDM). The stability of the scheme is analyzed. Convergence of the suggested method is proved. It is shown that the method is of order O ( Δ t 2 + h 4 ) . Some numerical experiments are carried out in order to illustrate the applicability and accuracy of the proposed method and to validate the theoretical results as well. It is shown that HOCFDM solution matches very well with the exact solution. Further, the rate of convergence predicted theoretically is the same as that obtained numerically. The computational time for our method in Matlab programming language is provided.

A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions

In this paper, we develop and analyze a high order compact finite difference method (CFDM) for solving a general class of two-point nonlinear singular boundary value problems with Neumann and Robin boundary conditions arising in various physical models. Convergence result of this method is established through matrix analysis approach. To illustrate the applicability and accuracy of the method, we consider nine numerical examples, including heat conduction in the human head, equilibrium of isothermal gas sphere, oxygen-diffusion in a spherical cell and reaction–diffusion process in a spherical permeable catalyst. It is shown that the computational order of convergence of the proposed CFDM is four. The obtained results are compared with those obtained by other existing numerical methods.

A high order scheme for unsteady heat conduction equations

In this work a high order scheme is proposed for solving the heat conduction problems. In this scheme, the governing equation of heat conduction is firstly written into a system of first-order partial differential equations which are integrated over the control volumes around each node point and a finite time step. The resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter. The error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible. Both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method. And the high order scheme demonstrates a better performance than the high order compact finite difference method and the Runge–Kutta discontinuous Galerkin method.

Fourth Order Compact Finite Difference Method for Solving One Dimensional Wave Equation

This paper introduces the fourth order compact finite difference method for solving the numerical solution of one-dimensional wave equations. The convergence of the method for the problem under consideration had been investigated. To validate the applicability of the method on the proposed equation, two model examples have been solved for different values of mesh sizes. The numerical results in terms of point wise absolute errors presented in tables and graphs show that the present method approximates the exact solution very well .

Study of optical modal index for semi conductor rib wave guides using higher order compact finite difference method

In this paper, a higher order compact finite difference method in combination with conjugate gradient iteration scheme has been used for optical modal index study of four different semi conductor rib wave guide structures in GaAs/GaAlAs and GeSi material systems. Obtained results for modal indices and normalized indices are used to optimize the structure parameters of the wave guides under consideration and confirm the degree of guidance for propagating waves through various structures in line with the results as obtained by other researchers. Confinement factors for different structures are calculated in support of the data to verify the acceptability of our method.

Fourth order compact finite difference method for solving singularly perturbed 1D reaction diffusion equations with dirichlet boundary conditions

A numerical method based on finite difference scheme with uniform mesh is presented for solving singularly perturbed two-point boundary value problems of 1D reaction-diffusion equations. First, the derivatives of the given differential equation is replaced by the finite difference approximations and then, solved by using fourth order compact finite difference method by taking uniform mesh. To demonstrate the efficiency of the method, numerical illustrations have been given. Graphs are also depicted in support of the numerical results. Both the theoretical and computational rate of convergence of the method have been examined and found to be in agreement. As it can be observed from the numerical results presented in tables and graphs, the present method approximates the exact solution very well.Keywords: Singular perturbation, Compact finite difference method, Reaction diffusion.

Tenth Order Compact Finite Difference Method for Solving Singularly Perturbed 1D Reaction - Diffusion Equations

In this paper, tenth order compact finite difference method have been presented for solving singularly perturbed two-point boundary value problems of 1D reaction-diffusion equations. The derivatives in the given differential equation have been replaced by finite difference approximations and transformed to tri-diagonal system which can easily be solved by Discrete Invariant Imbedding algorithm. The theoretical error bounds have been established for the method. Three model examples have been considered to check the applicability of the proposed method. The numerical results presented in tables show that the present method approximates the exact solution very well.

Optical confinement study of different semi conductor rib waveguides using higher order compact finite difference method

Semi conductor optical wave guides with well defined refractive index profile and geometric shapes which help to understand the propagation properties are widely used in the field of integrated optics. Accordingly rib waveguides are considered, as this structure allows the greatest versatility in device design in which the basic concept employed is its compositional changes in the vertical direction where the three regions of the rib structure are divided into zones with specific refractive indices. As for these types of structures no precise analytic solution is obtained, different numerical techniques have been implemented for their study in terms of their modal analysis. So an accurate higher order compact (HOC) finite difference method (FDM) in combination with conjugate gradient method (CGM) is used here for the investigation of refractive index profile of GaAs and GeSi rib waveguide structure to show their effect on transmission properties of the guided wave using Helmholtz wave equation. The difference in surface and contour plots of the polarized E-field solution clearly shows the material dependence of transmitted waves. We have also studied the variation of optical confinement factor with a number of parameters to understand the propagation properties appropriately as per the requirement of various guided wave optoelectronic devices.

A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

Higher order compact (HOC) finite difference method (FDM) to study optical confinement through semiconductor rib wave guides

Higher order compact (HOC) finite difference method (FDM) to study optical confinement through semiconductor rib wave guides

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrödinger Equations with Multiply Components

Considering the coupled nonlinear Schrödinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations. Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

A time dependent two-dimensional numerical simulation of flow around a rectangular cylinder is conducted using a particular high order compact finite difference method. The current forth-order compact scheme is implemented for the simulation of flow around bluff bodies for the first time. This formulation offers a semiexplicit method for the solution of Navier-Stokes equations in stream function-vorticity form and based on a nine point difference scheme with uniform grid spacing. The main advantage of the suggested method is its fast convergence and high accuracy. The proposed numerical method combines the enhanced Fourni’e’s forth order scheme with a semi-explicit formulation. By this combination, very accurate results can be obtained with a relatively coarse mesh in a short time. Uniform grids are applied with this high performance algorithm, and the accuracy of the scheme is proved using the benchmark problem of incompressible laminar flow around a rectangular cylinder at medium and high Reynolds numbers. Current results have good agreement with other numerical and experimental values. Generally, it is shown that, using the high order finite difference method demonstrates fairly competitive results for the current problem.