Order h4 difference methods for a class of singular two space elliptic boundary value problems

In this article, we derive difference methods of O(h⁴) for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points. We obtain two sets of fouth-order difference methods; one in the absence of mixed derivatives, second when the coefficients of u_xy are not equal to zero and the coefficients of u_xx and u_yy are equal. There do not exist fourth-order schemes involving nine grid points for the general case. The method having two variables has been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes' model equations in polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson's equation in polar coordinates. Some numerical examples are provided to illustrate the fourth-order convergence of the proposed methods.     

Technical note: The numerical solution of the system of 3-D nonlinear elliptic equations with mixed derivatives and variable coefficients using fourth-order difference methods

In this article, we report two fourth-order difference methods for the numerical integration of the system of general 3-D nonlinear elliptic equations subject to Dirichlet boundary conditions on a uniform cubic grid. When the coefficients of u_xy, u_yz, and u_zx are not equal to zero and the coefficients of u_xx, u_yy, and u_zz are equal, we require 27 grid points; when the coefficients of u_xy, u_yz, and u_zx are equal to zero, we require only 19 grid points. The utility of the new methods is shown by testing the methods on various examples, including 3-D steady state viscous incompressible Navier–Stokes' model equations and Poisson's equation in polar coordinates, which confirm the accurate and oscillation-free solutions for large Reynolds numbers even in the vicinity of singularity. © 1995 John Wiley & Sons, Inc.     

High-order methods for elliptic equations with variable coefficients

In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two-dimensional, second-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth-order schemes.     

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge ɛ-uniformly at the rate of O(N ⁻²ln² N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum norm at the same rate of O(N ⁻⁴ln⁴ N).     

A high-order compact difference scheme for 2D Laplace and Poisson equations in non-uniform grid systems

In this study, a high-order compact scheme for 2D Laplace and Poisson equations under a non-uniform grid setting is developed. Based on the optimal difference method, a nine-point compact difference scheme is generated. Difference coefficients at each grid point and source term are derived. This is accomplished through the consideration of compatibility between the partial differential equation and its difference discretization. Theoretically, the proposed scheme has third- to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Two examples are provided to examine performance of the proposed scheme. Compared with the traditional five-point difference scheme, the proposed scheme can produce more accurate results with faster convergence. Another reference scheme with the same nine-point grid stencil is derived based on the five-point scheme. The two nine-point schemes have the same coefficients for each grid points; however, their coefficients for the source term are different. The overall accuracy level of the solution resulting from the proposed scheme is higher than that of the nine-point reference scheme. It is also indicated that the smoothness of grids has significant effects on accuracy and convergence of the solutions; efforts in optimizing the grid configuration and allocation can improve solution accuracy and efficiency. Consequently, with the proposed method, solution under the non-uniform grid setting with appropriate grid allocation would be more accurate than that under the uniform-grid manipulation, with the same number of grid points.     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.     

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.     

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.     

Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

We derive a fourth-order compact finite difference scheme for a two-dimensional elliptic problem with a mixed derivative and constant coefficients. We conduct experimental study on numerical solution of the problem discretized by the present compact scheme and the traditional second-order central difference scheme. We study the computed accuracy achieved by each scheme and the performance of the Gauss-Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from the difference schemes.     

A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) − α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use ‘conditions of continuity’ to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over [a, b]. The fourth order of the presented method is demonstrated computationally by two examples.     

Two difference schemes for the numerical solution of Maxwell’s equations as applied to extremely and super low frequency signal propagation in the Earth-ionosphere waveguide

Two explicit two-time-level difference schemes for the numerical solution of Maxwell’s equations are proposed to simulate propagation of small-amplitude extremely and super low frequency electromagnetic signals (200 Hz and lower) in the Earth-ionosphere waveguide with allowance for the tensor conductivity of the ionosphere. Both schemes rely on a new approach to time approximation, specifically, on Maxwell’s equations represented in integral form with respect to time. The spatial derivatives in both schemes are approximated to fourth-order accuracy. The first scheme uses field equations and is second-order accurate in time. The second scheme uses potential equations and is fourth-order accurate in time. Comparative test computations show that the schemes have a number of important advantages over those based on finite-difference approximations of time derivatives. Additionally, the potential scheme is shown to possess better properties than the field scheme.     

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.     

A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel

In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L₁ discrete formula and the Riemann–Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

Approximate Hermite quasi-interpolation

In this article, we derive approximate quasi-interpolants when the values of a function u and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants, which provide high-order approximations for solutions to elliptic differential equations with constant coefficients.     

Fourth-order finite difference method for three-dimensional elliptic equations with nonlinear first-derivative terms

We present a 19-point fourth-order finite difference method for the nonlinear second-order system of three-dimensional elliptic equations Au _xx + Bu _yy + Cu _zz = f , where A , B , C , are M × M diagonal matrices, on a cubic region R subject to the Dirichlet boundary conditions u (x, y, z) = u ⁽⁰⁾(x, y, z) on ?R. We establish, under appropriate conditions, O(h⁴) convergence of the difference method. Numerical examples are given to illustrate the method and its fourth-order convergence. © 1992 John Wiley & Sons, Inc.     

An algebraic multigrid method for Q2−Q1 mixed discretizations of the Navier–Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q ₂? Q ₁ mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q ₂? Q ₁ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN‐AMG) is utilized. EMIN‐AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.     

求解对流扩散反应方程的四阶混合紧致差分方法

针对一维对流扩散反应方程,基于对流扩散方程的四阶指数型紧致差分格式,以及一阶导数的四阶Padé公式,发展了一种高效求解对流扩散反应方程的混合型四阶紧致差分格式.数值实验结果验证了格式对于边界层问题或大雷诺数或大Pelect数的大梯度问题的求解的高精度和鲁棒性的优点.     

The accuracy of a discrete spectral problem with mixed derivatives

The accuracy of a difference spectral problem for a second-order elliptic equation with mixed derivatives and constant coefficients is estimated. In so doing, the fact that the eigenfunctions belong to the Sobolyev space W ₂ ² in a rectangle is used. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 59–66.