Reduced-Order Finite Difference Scheme

The major difference between the ROEFDI scheme and the POD-based existing reduced-order extrapolating approaches (see, e.g., [22-25]) consists in that the ROEFDI scheme here is implicit and unconditionally stable so that its theoretical analysis and numerical simulations need more technique than the existing those as mentioned above, which are conditionally stable explicit schemes. Especially, the ROEFDI scheme here only uses the classical FD solutions on the first very short time span to constitute the POD basis and build the ROEFDI scheme. Therefore, it not only holds the unconditionally stable merit of the implicit FD scheme, but also includes not repeated computation as References [22- 25]. Hence, it is the development and improvement over the existing those as mentioned above.

In generally, it is difficult to get an exact solution for the GKS equation because of its complex nonlinearity, thus, it is solved by numerical methods. Meanwhile,GKS equation is also an important model for testing various numerical algorithm. In recently, several types of numerical methods have been developed for numerical simulation of the KS equation and GKS equation. For example, Akrivis [4] pre- sented a Crank-Nicolson-type finite difference scheme for KS equation with periodic boundary conditions. Khater et al. [5] extended the Chebyshev spectral collocation method to solve GKS equation. Mittal et al. [1] implemented quantic B-spline collocation method to find numerical solution of KS equation. Uddin et al. [2] applied radial basis function based meshfree method for the solution of KS equation. Later, Dabboura et al. [6] used moving least squares meshfree method to solve the GKS equation. Lakestani et al. [3] proposed B-spline function to solve this equation. Lai et al. [7] and Otomo et al. [8] investigated KS equation by lattice Boltzmann method. Singh et al. [9] presented the high- order compact finite difference scheme to simulate the KS equation.

This article is organized as follows. In Section  we derive a classical FD scheme with fully second-order accuracy for NSIBEs and generate snapshots from the first few numer- ical solutions obtained from the classical FD scheme with fully second-order accuracy. In Section  we construct orthonormal POD bases from the elements of the snapshots by means of the POD technique and establish the POD-based reduced-order FD extrap- olating scheme with fully second-order accuracy and lower dimensions for NSIBEs. In Section , the error estimates of the reduced-order FD solutions obtained from the POD reduced-order FD extrapolating model are provided as guidance to choose the number of POD basis functions and renew the POD basis, and the algorithm implementation for the POD-based reduced-order FD extrapolating model. In Section , a numerical experiment is used to show that the numerical results are consistent with the theoretical conclusions and we validate that the POD-based reduced-order FD extrapolating model is feasible and efficient for solving NSIBEs. In Section  we provide the main conclusions and present discussions.

The fractional-order differential equations have become a research hot-spot in science and engineering in recent years (see, e.g., [1, 2]). Unfortunately, there are very few researches of numerical solutions for the standard Riesz space fractional-order sine-Gordon equation (see, e.g., [3–5]). However, the Riesz space fractional-order parabolic-type sine-Gordon equation (RSFOPTSGE) with the first-order time derivative is not studied. Therefore, this paper mainly focuses on the study of the Crank–Nicolson finite difference (CNFD) scheme for RSFOPTSGE with first-order time derivative.

piecewise uniform mesh, we show theoretically that the coefficient matrix after discretiza- tion is an M-matrix. This ensures that the scheme is maximum-norm stable for arbitrary parameter settings. Most remarkably, we have successfully obtained a sharp error estimate of the current scheme by applying the maximum principle to the discrete linear comple- mentarity problem in two mesh sets. It is proved that the scheme is first and second order convergent with respect to the time and spatial variables, respectively. Numerical results agree with the theoretical statement and indicate that our method is more accurate than the other existing methods.

In this paper, an explicit exact finite-difference scheme for the Huxley equation is presented based on the nonstandard finite-difference (NSFD) scheme. Afterwards, an NSFD scheme is proposed for the numerical solu- tion of the Huxley equation. The positivity and boundedness of the scheme is discussed. It is shown through analysis that the proposed scheme is consis- tent, stable, and convergence. The numerical results obtained by the NSFD scheme is compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD scheme.

This paper is organized as follows : In the next section , we elaborate some basic definitions and properties of the Grunwald-Letnikov (GL) approximation and provides a brief overview of the important features of the procedures for constructing NSFD schemes for systems of ODEs . In Section 3 , we introduce fractional-order into the model that describes HIV-1 infection of CD4 T-cells and also stability theorem and Routh-Hurwitz stability conditions are given for the local asymptotic stability of the fractional systems . In Section 4 , we will discuss the stability analysis of fractional system . In Section 5 , we present the idea of NSFD scheme for solving the fractional-order HIV-1 infection of CD4 T-cells model . Finally in the last section , numerical results demonstrate that the NSFD approach is easy to be implemented and accurate when applied to the fractional-order HIV1 infection model.

In this paper, we give a difference scheme for the generalized Novikov equation. In Section 2, we give some preparation knowledge. In Section 3, we propose a conservative finite difference scheme for the generalized Novikov equation and use Brouwer fixed point theorem to obtain the existence of the solution for the corresponding difference equation. In Section 4, we prove the convergence and stability of the solution by using the discrete energy method.

in which the primed variables denote the respective quantities with dimensions. The Alfvén number β is a dimensionless quantity characterizing the flow in presence of magnetic field and it is the ratio of the speed of the Alfvén wave to the speed of the main stream fluid. The last term in Equation (2) is the body force term which is nonlinearly coupled with other equations in the list. Equation (1) is due to incompressibility condition while Equations ((3) and (4)) are the Maxwell’s equations to be satisfied for the applied magnetic field H . In the Amperes law (4), the second term is due to electric displacement vector D . Since displacement current density is negligible in fluid flows the second term in the right hand side of Equation (4) can be dropped. For steady state conditions, the electrodynamic continuity equation is (5) and the conduction current density j has to satisfy the Ohms law which is Equation (6). First it is noted that Equations ((1) to (6)) are coupled. Now, if heat transfer analysis is to be carried out, then the energy transport equation to be solved is given by Equation (7). If we solve (7) by treating the velocity q as coupled with Equations ((1) to (6)) then we can get the natural convection heat transfer properties. On the other hand, if we first solve the set of Equations ((1) to (6)) and provide this solution to Equation (7) as its input, then we can study the forced convection properties. In any case, first we need a discretization scheme to solve the governing equations. In the following, we propose to device a solution method based on streamfunction-vorticity approach and which is suitable for two- dimensional flow simulations. In cylindrical polar system, the velocity and applied magnetic field are

( < α < ) for fractional subdiffusion equation with spatially variable coefficient, whereas Wang et al. [] proposed a Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations and proved the well-posedness and optimal-order conver- gence of this method. Chen et al. [] presented a fast semiimplicit difference method with convergence order O(τ + h) for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients and also developed a fast accurate iterative method by decomposing the dense coefficient matrix into a combination of Toeplitz-like matri- ces, whereas Wang [] established a compact finite difference method with convergence order O(τ –α + τ  + h  ) ( < α < ) for a class of time fractional convection-diffusion-wave

of 5 MHz as source. The normalized time step ∆t c is equal to 0.5. The reference cell size ∆z and reference time step ∆t are 55.189 µm and 20.2157 ns, respectively. The spatial resolution, number of grid cells per wavelength, is taken as 9.8933. The motion-free boundary condition is used to trun- cate the simulation on all sides. A normal force is applied within a finite aperture at the top of the medium. The finite aperture is shown in red color. The EFIT and FDM snapshot of the wave field is taken at t = 90.9707 µs. In this case, us- ing FDM, accuracy is increased by a factor of 30 compared to EFIT.

It is non-less important that one can obtain lower estimates of the solutions to differential – difference problems, or in the general case, two-sided estimates for the solution of the problems. This is especially important for investigation of theoretical properties of the computational methods approximating problems with unbounded nonlinearities, where it is necessary to prove that discrete solution belongs to a neighbourhood of the exact solution. As an example we investigate the Gamma equation modelling pricing of options in financial mathematics. In this context, it is interesting to note the paper [11], in which two-sided estimates for solution of difference schemes approximating Dirichlet problem for linear parabolic equation are obtained in the discrete and continuous cases.

In 1D applications, a prescribed clustering or stretching function is used to map the unequally spaced nodes with physical x-coordinates to equally spaced computational ξ- coordinates. These 1D mapping functions can also be used for Cartesian meshes in 2D and 3D. The main motivation for the development of the Cell-Centred Finite Difference scheme was to obtain a finite difference formulation that could handle an arbitrary distribution of nodes in a mesh without having to introduce coordinate transformations. Despite its success for unstructured meshes, simple interpolation schemes may cause the numerical scheme to lose consistency. In order to overcome this problem, more complicated interpolations are needed, making the method less attractive for higher dimensional problems.

The above finite element scheme is used here to get the temperature and radial displacement through the ra- dial direction of the FGM hollow cylinder. The least square method is used also to get the appropriate stresses in the FGM hollow cylinder. The results are presented in the non -dimensional form:

We discretise the spatial β-order fractional derivative using the Gr¨unwald finite difference formula at all time levels. The standard Gr¨unwald estimate generally yields unstable finite difference equation regardless of whatever result in finite difference method is an explicit or an implicit system for related discussion [6], [16], [18]. Therefore, we use a right shifted Gr¨unwald formula to estimate the spatial β-order

In this present paper, we analyze the numerical solu- tion of the initial/boundary problems (1)-(4). The nu- merical method presented here comprises a fitted differ- ence scheme on an uniform mesh. Fitted operator method is widly used to construct and analyse uniform difference methods, especially for a linear differential problems (see, e.g., [4-7]). In the Section 2, we state some important properties of the exact solution. The derivation of the difference scheme and uniform convergence analysis have been given in Section 3. Uniform convergence is proved in the discrete maximum norm. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [8,9].

In this paper, we study the fractional-order Lotka-Volterra model. The sta- bility of equilibrium points is studied. Numerical solutions of these models are given. The reason for considering a fractional order system instead of its integer order counterpart is that fractional order differential equations are generalizations of integer order differential equations. Also using fractional order differential equations can help us to reduce the errors arising from the neglected parameters in modelling real life phenomena.

In the existing papers, many scholars focus their eyes on the compact finite difference scheme (CFDS), which has been widely used in numerical solution of various types of par- tial differential equations. The outstanding advantage of CFDS is that it possesses a faster convergence rate than the corresponding explicit schemes, not significantly increasing the points in each coordinate direction of the grid. As one of the most effective numerical im- plementations, there have been many numerical research reports concerning with CFDS. For example, Hammad [1] constructed CFDS for Burgers–Huxley and Burgers–Fisher equations. Wang [2] developed CFDS for Poisson equation. Mohebbi [3] combined CFDS with a radial basis functions meshless approach to solve the 2D Rayleigh–Stokes problem. Düring [4] applied CFDS to stochastic volatility models on non-uniform grids. Chen [5] provided high-order CFDS to solve parabolic equation. Especially, some attempts have been made whose main idea is to combine fourth Runge–Kutta in time and a sixth-order compact finite difference in space (CFDS6) by the researchers [6–8]. However, the CFDS6 for parabolic equation, especially the case of desirable accuracy in high dimension, they usually need small spatial discretization or extended finite difference stencils and a small time step which brings about heavy computational loads. Therefore, an important prob- lem for CFDS6 is how to build a scheme which not only saves the computational time in the practical problems but also holds a sufficiently accurate numerical solution.

As a result, the SFDTD(3,4) scheme is constructed. The Courant- Friedrichs-Levy (CFL) number of the SFDTD(3,4) scheme is 1.118, which is bigger than 0.743 for the SFDTD(4,4) scheme proposed in [17]. In each time step, the SFDTD(3,4) scheme only requires three stages compared with five stages for the SFDTD(4,4) scheme. Hence, considerable CPU time will be saved by the three-stage third-order symplectic integrators. Figure 1 gives the relative phase velocity error as a function of points per wavelength (PPW) for a plane wave traveling at θ = 60 ◦ and φ = 30 ◦ . The CFL number is set to be 0.495 that is

This note has analyzed a finite difference scheme which approximates a second order differential equation. It has been shown to be zero-stable (in the sense of Dahlquist) and convergent. The fact that this scheme is zero-stable means that consistency, itself, implies convergence. Although this work has treated a simple finite difference scheme, the ideas are quite general and may, in principle, be applied to demonstrate the convergence of other more sophisticated schemes.