Fourth-order Finite Difference Approximation Research Articles

On a sparse and stable solver on graded meshes for solving high-dimensional parabolic pricing PDEs

The purpose of this work is to investigate the multi-dimensional Black-Scholes partial differential equation with variable coefficients numerically. The problem is of practical importance due to option pricing at the presence of multi assets. Since by increasing the dimension, the curse of dimensionality restricts the computations, the proposed solver will be constructed based on sparse arrays. Toward this goal, fourth-order finite difference approximations on graded meshes are introduced and then employed through semi-discretization. Then a sixth-order Runge-Kutta solver is employed for finding the resolution of the derived set of ordinary differential equations. The stability of the proposed scheme is furnished in detail as well. Numerical testings are given to uphold the accuracy and efficacy of the proposed procedure.

MHD double-diffusive mixed convection and entropy generation of nanofluid in a trapezoidal cavity

Double diffusive, Magnetohydrodynamics (MHD), mixed convection flow of Al2O3-water nanofluid inside a trapezoidal enclosure with discrete heat and mass source at the bottom wall of the cavity has been discussed numerically. Together with these entropy generations due to fluid friction, heat transfer, mass transfer and magnetic field are discussed. The upper wall of the cavity is considered to move with a fixed velocity U0 towards positive X-direction. Different inclination angles and different aspect ratios are considered in the present study. Second and fourth order finite difference approximations are used to solve the governing equations by using biconjugate gradient stabilized method (BiCGStab). It is observed that, entropy generation mostly depends on heat transfer and lower values of Richardson number (Ri=0.01) reduces the entropy generation. The highest quantitative value of |Sθ|max is 21,664 for Ri=100,Re=100andϕ=450 and lowest quantative value of |Sθ|max is 5,824 for Ri=0.01,Re=1000andϕ=600 throughout the present study. Also the highest quantitative values of average Nusselt and Sherwood numbers are 0.2597 and 0.4208 for Ri=10,Re=100andϕ=45∘. The main objective of our present study is to minimize entropy generations due to combined effects of fluid flow, heat transfer, mass transfer and the effect of magnetic field so that the loss of energy can be reduced. This study observed that mixed convection flow, low magnetic field and low aspect ratio cavity of rectangular shape is always preferable to reduce total entropy generation and the middle part of the lower wall of the cavity is highly acceptable for heat and mass source for sprinkle of heat and mass throughout the whole of the cavity.

Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

A Low-Dispersion Time Differencing Scheme for Maxwell’s Equations

A novel low-dispersion time differencing scheme for the integration of Maxwell’s equations is developed. The method employs a staggered temporal grid with a time difference operator that controls the magnitude of dispersion errors. It is tested within the physical domains employing fourth-order finite difference approximations but can be combined with grids using any order of accuracy, including the spectral grids. Stability analysis demonstrates that the proposed method achieves third- and fourth-order accuracies for amplitude and phase errors, respectively, outperforming the second-order phase error produced by the finite-difference time-domain methods based on the Yee scheme. The order of accuracy accomplished by the method and its zone of stability are comparable to the third-order Runge–Kutta scheme with the advantage of being three times more efficient. The proposed method is proven to be effective and stable when employed within the perfectly matched layers and nonlinear Kerr media. Its performance and ability to reduce the dispersion errors are demonstrated by comparing the solutions computed by the method with those obtained from the Yee and other high-order schemes.

Influence of unsteady pressure-flow conditions on strength of steel pipelines with volumetric defects reinforced by composite sleeves

Structural integrity and risk management have a wide interest because of its practical applications, such as oil and gas pipelines, piping systems under pressure in power stations, urban water, and heating networks. The main goal of this paper is twofold. Firstly, to estimate the unsteady pressureflow variations in a gas transmission grid within the framework of sequential data assimilation. This technique enables to determine accurately the maximum pressure at the localized defect on the pipeline by merging measurements that contain random errors into the inexact numerical flow model. For this purpose, a particle filter is used. The semi-discretization approach is applied to convert the nonisothermal flow model into an initial value problem of ordinary differential equations. The spatial discretization is based on a five-point, fourth-order finite difference approximation and the time marching was done using a diagonally implicit Runge-Kutta scheme. Secondly, to study the strength of steel tubes reinforced with composite sleeves containing localized part-wall thickness losses caused by corrosion while taking into consideration a safe operating pressure. For a steel thin-walled cylinder containing a wrap of fiberglass with epoxy resin, the burst pressure and sleeve thickness are determined. Finally, the repaired pipeline with a fiber-reinforced composite sleeve is investigated. The results enable operators to handle problems of corroded steel pipelines and develop effective repair activities during operation. For this reason, current research is important for the maintenance of underground steel networks.

Pseudo-impulsive solutions of the forward-speed diffraction problem using a high-order finite-difference method

This paper considers pseudo-impulsive numerical solutions to the forward-speed diffraction problem, as derived from classical linearized potential flow theory. Both head- and following-seas cases are treated. Fourth-order finite-difference approximations are applied on overlapping, boundary-fitted grids to obtain solutions using both the Neumann-Kelvin and the double-body flow linearizations of the problem. A method for computing the pseudo-impulsive incident wave forcing in finite water depth using the Fast Fourier Transform (FFT) is presented. The pseudo-impulsive scattering solution is then Fourier transformed into the frequency domain to obtain the wave excitation forces and the body motion response. The calculations are validated against reference solutions for a submerged circular cylinder and a submerged sphere. Calculations are also made for a modern bulk carrier, showing good agreement with experimental measurements.

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas–Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

Comparison of Bayesian estimation methods for modeling flow transients in gas pipelines

The aim of this work was to investigate the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and sequential importance resampling (SIR) filter for real-time estimation of a hyperbolic partial differential equation (PDE) system describing the gas flow transients in pipelines. The numerical method of lines was used for solving the system of PDEs. The spatial discretization was based on a five-point, fourth-order finite difference approximation and the time integrations were done using the fourth-order Runge-Kutta scheme (RK4). The resulting nonlinear state-space model was used by the Bayesian filtering algorithms for estimation of the system states. Numerical experiments were conducted to compare the accuracy, robustness and computation time. The results indicated that the difference in terms of accuracy between the EKF and UKF was small and considering the computation time of the UKF and numerically robustness of calculating the Jacobian matrix, the use of EKF might be a better choice when fast solutions are required. The UKF outperformed the EKF and SIR filter when steep variations in the mass flow rate boundary conditions occurred together with low model uncertainty.

Time-filtered leapfrog integration of Maxwell equations using unstaggered temporal grids

A finite-difference time-domain method for integration of Maxwell equations is presented. The computational algorithm is based on the leapfrog time stepping scheme with unstaggered temporal grids. It uses a fourth-order implicit time filter that reduces computational modes and fourth-order finite difference approximations for spatial derivatives. The method can be applied within both staggered and collocated spatial grids. It has the advantage of allowing explicit treatment of terms involving electric current density and application of selective numerical smoothing which can be used to smooth out errors generated by finite differencing. In addition, the method does not require iteration of the electric constitutive relation in nonlinear electromagnetic propagation problems. The numerical method is shown to be effective and stable when employed within Perfectly Matched Layers (PML). Stability analysis demonstrates that the proposed method is effective in stabilizing and controlling numerical instabilities of computational modes arising in wave propagation problems with physical damping and artificial smoothing terms while maintaining higher accuracy for the physical modes. Comparison of simulation results obtained from the proposed method and those computed by the classical time filtered leapfrog, where Maxwell equations are integrated for a lossy medium, within PML regions and for Kerr-nonlinear media show that the proposed method is robust and accurate. The performance of the computational algorithm is also verified by analyzing parametric four wave mixing in an optical nonlinear Kerr medium. The algorithm is found to accurately predict frequencies and amplitudes of nonlinearly converted waves under realistic conditions proposed in the literature.

An Efficient Direct Method to Solve the Three Dimensional Poisson’s Equation

In this work, the three dimensional Poisson’s equation in Cartesian coordinates with the Dirichlet’s boundary conditions in a cube is solved directly, by extending the method of Hockney. The Poisson equation is approximated by 19-points and 27-points fourth order finite difference approximation schemes and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The efficiency of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. It is shown that 19-point formula produces comparable results with 27-point formula, though computational efforts are more in 27-point formula.

Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System

In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invokes also a sixth order Pade approximation for the first order derivatives, using an approximation suggested by Carpenter-Gottlieb-Abarbanel (J. Comput. Phys. 108:272---295, 1993). We also introduce the derivation of a pure streamfunction formulation for the Navier-Stokes equations in three dimensions.

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u ( x , y , z ) and its Laplacian ∇ 2 u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.

A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach

AbstractIn this article, using coupled approach, we discuss fourth order finite difference approximation for the solution of two dimensional nonlinear biharmonic partial differential equations on a 9‐point compact stencil. The solutions of unknown variable and its Laplacian are obtained at each internal grid points. This discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions. We require only system of two equations to obtain the solution and its Laplacian. The proposed fourth order method is used to solve a set of test problems and produce high accuracy numerical solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind

In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions.

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.

A family of fourth‐order parallel splitting methods for parabolic partial differential equations

A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial dderivatives are approximated by fourth-order finite-difference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional head equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 357–373, 1997

The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension

We construct local fourth-order finite difference approximations of first and second derivatives, on nonuniform grids, in one dimension. The approximations are required to satisfy symmetry relationships that come from the analogous higher-dimensional fundamental operators: the divergence, the gradient, and the Laplacian. For example, we require that the discrete divergence and gradient be negative adjoint of each other, DIV∗ = − GRAD , and the discrete Laplacian is defined as LAP = DIVGRAD . The adjointness requirement on the divergence and gradient guarantees that the Laplacian is a symmetric negative operator. The discrete approximations we derive are fourth-order on smooth grids, but the approach can be extended to create approximations of arbitrarily high order. We analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids.

Comments on “Truncation Errors in Finite-Difference Estimates of Geostrophic Wind and Relative Vorticity”

Abstract Peppler and Smith (1984) discussed truncation errors associated with second-order and fourth-order finite difference approximations used to calculate the geostrophic wind and relative vorticity. They found that these errors were, in general, smaller for longer wavelengths, finer-grid resolution, and fourth-order differencing. Second-order differencing produced fields that were numerically less than their corresponding analytic values and yielded errors which decreased with reduced grid interval. Fourth-order differencing decreased the errors when the grid interval was reduced, but only while the wavelength was ten limes or more greater than the grid interval. Results presented here indicate that the wind speed and Vorticity errors estimated by the fourth-order scheme decreased when the grid interval was decreased independent of wavelength. Comparison with Peppler and Smith's actual computations showed only one difference: in their finite-(difference equations the coefficients were truncated to th...

The numerical solution of non-linear partial differential equations by the method of lines

A new method for solving non-linear parabolic partial differential equations, based on the method of lines, is developed. Second order and fourth order finite difference approximations to the spatial derivatives are used, and in each case the band structure of the associated Jacobian is exploited. This, together with the use of Gear's fourth order stiffly stable method in Nordsieck form, leads to a method which compares favourably with the respected Sincovec and Madsen method on Burgers' equation. The method has been tested on a number of difficult problems in the literature and has proved to be most successful.

NUMERICAL ADVECTION EXPERIMENTS1

Abstract First, second, and fourth order finite difference approximations to the color equation in both advection and conservation form are considered in one and two space dimensions. All schemes considered are based on forward time differences and most involve centered space differences. All are shown to be numerically stable for |uΔt/Δx| ≤ 1. Test calculations indicate that for the same order of accuracy, the conservation form produces more accurate solutions than the advection form. For either conservation or advection form, fourth order schemes are shown to be more accurate than second or first order schemes in terms of both amplitude and phase errors.