Standard Finite Difference Schemes Research Articles

High-order implicit finite difference schemes for the two-dimensional Poisson equation

In this paper, a new family of high-order finite difference schemes is proposed to solve the two-dimensional Poisson equation by implicit finite difference formulas of (2M+1) operator points. The implicit formulation is obtained from Taylor series expansion and wave plane theory analysis, and it is constructed from a few modifications to the standard finite difference schemes. The approximations achieve (2M+4)-order accuracy for the inner grid points and up to eighth-order accuracy for the boundary grid points. Using a Successive Over-Relaxation method, the high-order implicit schemes have faster convergence as M is increased, compensating the additional computation of more operator points. Thus, the proposed solver results in an attractive method, easy to implement, with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation. In addition, particular case M=1 yields a new compact finite difference schemes of sixth-order of accuracy. Numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes.

An energy-preserving discretization for the Poisson–Nernst–Planck equations

The Poisson---Nernst---Planck (PNP) equations have recently been used to describe the dynamics of ion transport through biological ion channels besides being widely employed in semiconductor industry. This paper is about the design of a numerical scheme to solve the PNP equations that preserves exactly (up to roundoff error) a discretized form of the energy dynamics of the system. The proposed finite difference scheme is of second-order accurate in both space and time. Comparisons are made between this energy dynamics-preserving scheme and a standard finite difference scheme, showing a difference in satisfying the energy law. Numerical results are presented for validating the orders of convergence in both time and space of the new scheme for the PNP system.

Revisiting the spectral analysis for high-order spectral discontinuous methods

The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion-relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space–time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,π], following the Matrix Power method. For wavenumbers above π, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than π. Since the new criteria recover the previous results for wavenumbers below π, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes.

A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms

In this article, a parameter-uniform hybrid numerical method is presented to solve a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms. Due to these discontinuities, interior layers appear in the solution of the problem considered. The hybrid numerical method uses the standard finite difference scheme in the coarse mesh region and the cubic spline difference scheme in the fine mesh region which is constructed on piecewise-uniform Shishkin mesh. Second order one sided difference approximations are used at the point of discontinuity. Error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to validate the theoretical results.

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton\u2013Jacobi\u2013Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton–Jacobi–Bellman equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.

An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.

A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition

A Robin type boundary value problem for a singularly perturbed parabolic delay differential equation is studied on a rectangular domain in the x - t plane. The second-order space derivative is multiplied by a small parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference scheme on a rectangular piecewise uniform fitted mesh of Nx × Nt elements condensing in the boundary layers is suggested and it is proved to be parameter-uniform. More specifically, it is shown that the errors are bounded in the maximum norm by C(Nx−2ln2Nx+Nt−1), where C is a constant independent of Nx, Nt and the small parameter. To validate the theoretical result an example is provided.

Discrete and continuous fractional persistence problems – the positivity property and applications

In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical (α=1) differential equation in the fractional case (here using fractional Caputo’s derivatives) and the numerical scheme which are associated (here with discrete Grünwald–Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Grünwald–Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey–predator model introduced by Javidi et al.

Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations

A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter e (e ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not e-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.

Numerical Study of an Initial-Boundary Value Neumann Problem for a Singularly Perturbed Parabolic Equation

For a singularly perturbed one-dimensional parabolic equation with a perturbation parameter e multiplying the highest-order derivative in the equation, e ∈ (0,1], an initial-boundary value Neumann problem is considered on a segment. In this problem, when the parameter e tends to zero, boundary layers appear in neighborhoods of the lateral boundary. In the paper, convergence of the problem solution and its regular and singular components are studied. It is shown that standard finite difference schemes on uniform grids used to numerically solve this problem do not converge e -uniformly. The error in the grid solution grows unboundedly when the parameter e → 0. The use of a special difference scheme on Shishkin grid which is a piecewise-uniform mesh with respect to x condensing in neighborhoods of boundary layers and a uniform mesh in t , constructed by using monotone grid approximations of the differential problems, allows us to find a numerical solution of this problem convergent in the maximum norm e -uniformly. The results of the numerical experiments confirm the theoretical results.

Simulations of the Helmholtz equation at any wave number for adaptive grids using a modified central finite difference scheme

In this paper, a modified central finite difference scheme for a three-point nonuniform grid is presented for the one-dimensional homogeneous Helmholtz equation using the Bloch wave property. The modified scheme provides highly accurate solutions at the nodes of the nonuniform grid for very small to very large range of wave numbers irrespective of how the grid is adapted throughout the domain. A variety of numerical examples are considered to validate the superiority of the modified scheme for a nonuniform grid over a standard central finite difference scheme.

An Effective Computational Scheme for the Optimal Control of Wave Equations

Abstract: We proposed and analyzed a new leapfrog finite difference scheme in time for solving the first-order necessary optimality systems arising in optimal control of wave equations. With a standard second-order central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy. Moreover, based on its favorable structure, an efficient preconditioned iterative method is provided for solving the discretized unsymmetric sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of our proposed algorithms.

Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard equation.

We investigate microphase separation patterns on curved surfaces in three-dimensional space by numerically solving a nonlocal Cahn-Hilliard equation for diblock copolymers. In our model, a curved surface is implicitly represented as the zero level set of a signed distance function. We employ a discrete narrow band grid that neighbors the curved surface. Using the closest point method, we apply a pseudo-Neumann boundary at the boundary of the computational domain. The boundary treatment allows us to replace the Laplace-Beltrami operator by the standard Laplacian operator. In particular, we can apply standard finite difference schemes in order to approximate the nonlocal Cahn-Hilliard equation in the discrete narrow band domain. We employ a type of unconditionally stable scheme, which was introduced by Eyre, and use the Jacobi iterative to solve the resulting implicit discrete system of equations. In addition, we use the minimum number of grid points for the discrete narrow band domain. Therefore, the algorithm is simple and fast. Numerous computational experiments are provided to study microphase separation patterns for diblock copolymers on curved surfaces in three-dimensional space.

Motion by mean curvature of curves on surfaces using the Allen–Cahn equation

In this paper we develop a fast and accurate numerical method for motion by mean curvature of curves on a surface in three-dimensional space using the Allen–Cahn equation. We use a narrow band domain. We adopt a hybrid explicit numerical method which is based on an operator splitting method. First, we solve the heat equation by using an explicit standard Cartesian finite difference scheme. For the domain boundary cells, we use an interpolation using the closest point method. Then, we update the solution by using a closed-form solution. The proposing numerical algorithm is computationally efficient since we use a hybrid explicit numerical scheme and solve the governing equation only on the narrow domain. We perform a series of numerical experiments. The computational results are consistent with known analytic solutions.

A new mimetic scheme for the acoustic wave equation

A new mimetic finite difference scheme for solving the acoustic wave equation is presented. It combines a novel second order tensor mimetic discretizations in space and a leapfrog approximation in time to produce an explicit multidimensional scheme. Convergence analysis of the new scheme on a staggered grid shows that it can take larger time steps than standard finite difference schemes based on ghost points formulation. A set of numerical test problems gives evidence of the versatility of the new mimetic scheme for handling general boundary conditions.

Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids

A diffusion driven model for hepatitis B virus (HBV) infection, taking into account the spatial mobility of both the HBV and the HBV DNA-containing capsids is presented. The global stability for the continuous model is discussed in terms of the basic reproduction number. The analysis is further carried out on a discretized version of the model. Since the standard finite difference (SFD) approximation could potentially lead to numerical instability, it has to be restricted or eliminated through dynamic consistency. The latter is accomplished by using a non-standard finite difference (NSFD) scheme and the global stability properties of the discretized model are studied. The results are numerically illustrated for the dynamics and stability of the various populations in addition to demonstrating the advantages of the usage of NSFD method over the SFD scheme.

An explicit nonstandard finite difference scheme for the Allen–Cahn equation

We design explicit nonstandard finite difference schemes for the nonlinear Allen–Cahn reaction diffusion equation in the limit of very small interaction length . In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space-independent and the time-independent different models. An exact nonstandard scheme is proposed for the space-independent model and energy conservative schemes are proposed for the time-independent model. We show the power of the derived scheme over the existing schemes through several numerical examples.

A fast direct solver for a fourth order finite difference scheme for Poisson’s equation on the unit disc in polar coordinates

We present a fourth order finite difference scheme for solving Poisson's equation on the unit disc in polar coordinates. We use a half-point shift in the r direction to avoid approximating the solution at r = 0. We derive our scheme from analysis of the local truncation error of the standard second order finite difference scheme. The resulting linear system is solved very efficiently (with cost almost proportional to the number of unknowns) using a matrix decomposition algorithm with fast Fourier transforms.

Second order method for solving 3D elasticity equations with complex interfaces

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three-dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equations. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques have been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuities of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins.

Compact Schemes for the Simulations of Wave Propagations

In this paper, a general frame work for the development of compact schemes in particular for time harmonic wave equation is presented. The salient features this frame work offers are (a) exact values of numerical solutions at the nodes of the spatial grid irrespective of one or higher dimensions are obtained; (b) compact schemes preserves same stencil structure as that of the standard finite difference and finite element schemes; (c) requirement of fine mesh size to enjoy desired level of accuracy is removed which is real trouble in the case of high wave numbers. Keywords: Time harmonic wave equation; High wave number; Compact schemes; Wave propagations. 2010 Mathematics Subject Classification: 65N15, 65N30, 65N35..