On some high accuracy difference schemes for solving elliptic equations

Summary Two well known high accuracy Alternating Direction Implicit difference schemes for solving Laplace's equation and the Biharmonic equation are considered. The set of iteration parameters of Douglas is used in both problems. More complete optimum values of the parameters involved are given.     

High-order splitting schemes for semilinear evolution equations

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.     

Study of convergence of difference schemes for quasilinear elliptic equations with degeneracy

Translated from Issledovaniya po Prikladnoi Matematike, No. 2, pp. 81–84, 1974.     

High-order accurate difference schemes for solving gasdynamic equations by the Godunov method with antidiffusion

A technique is proposed for improving the accuracy of the Godunov method as applied to gasdynamic simulations in one dimension. The underlying idea is the reconstruction of fluxes arsoss cell boundaries (“large” values) by using antidiffusion corrections, which are obtained by analyzing the differential approximation of the schemes. In contrast to other approaches, the reconstructed values are not the initial data but rather large values calculated by solving the Riemann problem. The approach is efficient and yields higher accuracy difference schemes with a high resolution.     

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h⁴. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

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.     

Convergence of centered difference schemes for a system of two-dimensional equations of acoustics

The accuracy of difference schemes for first-order hyperbolic systems is studied for the case of two-dimensional equations of acoustics with various boundary conditions. A difference scheme is constructed and an a priori bound of the error is obtained in some weak norm. This bound combined with the Bramble-Hilbert theorem makes it possible to prove o(^m + h^m) convergence of the difference solution to the solution of the differential problem in the class W₂ ^m(Q_T, m=1,2.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 48–57, 1985.     

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.     

High-order finite difference schemes for the solution of second-order BVPs

We introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh.     

Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations

In this paper, alternating direction implicit compact finite difference schemes are devised for the numerical solution of two-dimensional Schrödinger equations. The convergence rates of the present schemes are of order O(h⁴+τ²). Numerical experiments show that these schemes preserve the conservation laws of charge and energy and achieve the expected convergence rates. Representative simulations show that the proposed schemes are applicable to problems of engineering interest and competitive when compared to other existing procedures.     

High-order multigrid strategies for hybrid high-order discretizations of elliptic equations

This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high-order method. Combinations of $h$-, $p$-, and $hp$-coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non-nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.     

Defect correction for nonlinear elliptic difference equations

Summary The present paper is concerned with the study of a high-order defect correction technique for discretizations of nonlinear elliptic boundary value problems. The convergence of the method is analyzed in general and, in more detail, for a specific example. The algorithmic combination of defect correction and multigrid techniques is also discussed.This work was supported by Österreichischer Fonds zur Förderung der wissenschaftlichen Forschung     

分数阶扩散方程的隐差分格式

根据移位的Grnwald方法,得到求解分数阶扩散方程的三类隐差分格式.利用分数阶von Neumann方法,证明了求解亚扩散方程的两类差分格式是无条件稳定的,而求解超扩散方程的差分格式是条件稳定的,同时也给出了相应差分格式的局部截断误差估计.最后,通过两个数值例子证实了所提出的差分格式的正确性和有效性.     

Stable difference schemes for certain parabolic equations

In some applications, boundary value problems for second-order parabolic equations with a special nonself-adjoint operator have to be solved approximately. The operator of such a problem is a weighted sum of self-adjoint elliptic operators. Unconditionally stable two-level schemes are constructed taking into account that the operator of the problem is not self-adjoint. The possibilities of using explicit-implicit approximations in time and introducing a new sought variable are discussed. Splitting schemes are constructed whose numerical implementation involves the solution of auxiliary problems with self-adjoint operators.     

Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type

Summary High accuracy alternating direction implicit difference schemes for the heat equation, LAPLACE's equation and the biharmonic equation are considered. In addition to surveying the existing methods, several new methods are introduced. Sequences of iteration parameters are obtained for the elliptic problems and a numerical example is given.