Compact Finite Difference Approximation Research Articles

A FOURTH ORDER ENERGY STABLE FINITE DIFFERENCE SCHEME FOR A TIMOSHENKO BEAM EQUATIONS WITH LOCALLY DISTRIBUTED FEEDBACK

This work deals with the numerical solution of a control problem governed by the Timoshenko beam equations with locally distributed feedback. We apply a fourth-order Compact Finite Difference (CFD) approximation for the discretizing spatial derivatives and a Forward second order method for the resulting linear system of ordinary differential equations. Using the energy method, we derive energy relation for the continuous model, and design numerical scheme that preserve a discrete analogue of the energy relation. Numerical results show that the CFD approximation of fourth order give an efficient method for solving the Timoshenko beam equations.

Wavelet-optimized compact finite difference method for convection–diffusion equations

Abstract In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

MAGNETIC FIELD EFFECT ON THE HEAT TRANSFER IN A NANOFLUID FILLED LID DRIVEN CAVITY WITH JOULE HEATING

In this paper, the effects of magnetic field, Joule heating and volumetric heat generation on the heat transfer and fluid flow in a Cu-Water nanofluid filled lid driven cavity using enhanced streamfunction–velocity method are investigated. The cavity is heated by a uniform volumetric heat density and side walls have constant temperature. The top wall moves with constant velocity in +x direction, while no-slip boundary conditions are imposed on the other walls of the cavity. An inclined fixed magnetic field is applied to the left side wall of the cavity. The dimensionless governing equations are solved numerically for the stream function and temperature using finite difference method for various Richardson(Ri), Reynolds(Re), Hartmann (Ha), Eckert(Ec)numbers, magnetic field angle(α) and solid volume fraction of the nanofluid() in MATLAB software. To discretize the streamfunction-velocity formulation, a five point constant coefficient second-order compact finite difference approximation which avoids difficulties inherent in the conventional streamfunction–vorticity and primitive variable formulations is used. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB) and the temperature equation is solved using Jacobi bi-conjugate gradient stabilized (BiCGSTAB) method. The heat transfer within the cavity is characterized by Nusselt number (Nu1). The results show that Nu1 is significantly increased by increasing Ri and  and increasing the Reynolds number enhances convective cooling. The heat transfer within the cavity is decreased by increasing Hartmann number which improves conduction heat transfer and reduces Nu1. Joule heating has a negative effect on the convection within the cavity and convection is decreased by increasing the value of Ec. It can be investigated that Nu1 is decreased by increasing Ec due to the strong distortion effect of Joule heating on convection current of heat transfer.

Compact Finite Difference Schemes for Approximating Differential Relations

Differential relations include both differential operators and solvers for boundary value problems. The formulas of compact finite difference approximations for first- and second-order differential relations of the form $${{P}_{1}}[u] = {{P}_{2}}[f]$$ are obtained. An approximation on three-point stencils is used. Its implementation, as in the case of classical difference schemes, requires the inversion of a tridiagonal matrix. However, compact schemes provide significantly higher accuracy and the fourth order of approximation instead of the second.

Numerical Computation and Stability Analysis for the Fractional Subdiffusions with Spatial Variable Coefficients

In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.

Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.

An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Optimality and resonances in a class of compact finite difference schemes of high order

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Pade approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction.

Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients

ABSTRACTThis paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients. Based on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods.

A fourth-order extrapolated compact difference method for time-fractional convection-reaction-diffusion equations with spatially variable coefficients

This paper is concerned with numerical methods for a class of time-fractional convection-reaction-diffusion equations. The convection and reaction coefficients of the equation may be spatially variable. Based on the weighted and shifted Grünwald–Letnikov formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The local truncation error and the solvability of the resulting scheme are discussed in detail. The stability of the method and its convergence of third-order in time and fourth-order in space are rigorously proved by the discrete energy method. Combining this method with a Richardson extrapolation, we present an extrapolated compact difference method which is fourth-order accurate in both time and space. A rigorous proof for the convergence of the extrapolation method is given. Numerical results confirm our theoretical analysis, and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method.

Multigrid cell-centered techniques for high-order incompressible flow numerical solutions

A multigrid pressure correction scheme suitable for high order discretizations of the incompressible Navier–Stokes equations is developed and demonstrated. The pressure correction equation is discretized with fourth-order compact finite-difference approximations. Iterative methods based on multigrid techniques accelerate the most demanding part of the overall solution algorithm, which is the numerical solution of the arised large and sparse linear system. Geometrical multigrid methods, using partial semicoarsenig strategy and zebra line Gauss–Seidel relaxation, are employed to efficiently approximate the solution of the resulting algebraic linear system. Effects of various multigrid components on the pressure correction procedure are evaluated and new high-order transfer operators are developed for the case of cell-centered grids. Their convergence rates are also compared with commonly used intergrid transfer operators. Furthermore, numerically comparisons between different multigrid cycle approaches, such as V-, W- and F-cycle, are presented. The performance tests demonstrate that the new pressure correction approach significantly reduces the computational effort compared to single-grid algorithms. Furthermore, it is shown that the overall high order accuracy of the numerical method is retained in space and time with increasing Reynolds number.

Heat transfer enhancement via Görtler flow with spatial numerical simulation

PurposeThe centrifugal instability mechanism of boundary layers over concave surfaces is responsible for the development of quasi-periodic, counter-rotating vortices aligned in a streamwise direction known as Görtler vortices. By distorting the boundary layer structure in both the spanwise and the wall-normal directions, Görtler vortices may modify heat transfer rates. The purpose of this study is to conduct spatial numerical simulation experiments based on a vorticity–velocity formulation of the incompressible Navier–Stokes system of equations to quantify the role of the transition in the heat transfer process.Design/methodology/approachExperiments are conducted using an in-house, parallel, message-passing code. Compact finite difference approximations and a spectral method are used to approximate spatial derivatives. A fourth-order Runge–Kutta method is adopted for time integration. The Poisson equation is solved using a geometric multigrid method.FindingsResults show that the numerical method can capture the physics of transitional flows over concave geometries. They also show that the heat transfer rates in the late stages of the transition may be greater than those for either laminar or turbulent ones.Originality/valueThe numerical method can be considered as a robust alternative to investigate heat transfer properties in transitional boundary layer flows over concave surfaces.

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation.

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical “energy” of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

Fast Valuation of CEV American Options

The semi-analytical valuation of CEV American options is either computationally expensive when it delivers sufficiently accurate prices, lacks accuracy when computations are faster, or involves functions whose computations can be unstable. Trinomial schemes have the disadvantage that there are no built-in ways to compute hedging sensitivities and re-runs are required for the computation of deltas and gammas. We propose a faster computational technique by discretizing the no-arbitrage partial differential equation using a high-order three-point (compact) finite difference approximation combined with a simple extrapolation process in time. This new CEV American option pricing algorithm is simple to implement and is shown to yield prices with high accuracy. The superiority of the procedure over existing schemes is numerically demonstrated.

Numerical solution of Schrodinger equation using compact finite differences method and the cubic spline functions

In this paper, a high-order method for solving the Schrodinger equation is introduced. We apply a compact finite difference approximation for discretizing spatial derivatives and we use the C1-cubic spline collocation method for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. We can obtain both pointwise approximations at the all mesh points and, a cubic spline solution in each space step by the method. Numerical results show that the method is an efficient technique for solving the one-dimensional Schrodinger equation.Â

A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point

The problem of finding the solution of partial differential equation with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, chemical diffusion and control theory. In this paper, we use a high-order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem. In the proposed numerical scheme, we replace the space derivative with a fourth-order compact finite difference approximation. We will investigate the stability and convergence of proposed scheme and show that the convergence order is . Also due to the usually ill-posed nature of inverse problems, we examine the stability of method with respect to perturbations of the data. Numerical results corroborate the theoretical results and high accuracy of proposed scheme in comparison with the other methods in the literature.

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014