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.     

High-order difference schemes for two-dimensional elliptic equations

A high-order finite-difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h⁴. Several test problems, including one modeling convection-dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

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.     

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.     

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.     

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.     

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.     

On the spectral properties of three-layer difference schemes for parabolic equations with nonlocal conditions

We consider three-layer difference schemes for a one-dimensional linear parabolic equation with nonlocal integral conditions. A three-layer scheme is written out in an equivalent form of a two-layer scheme. We analyze the dependence of the spectrum of the difference operator on the parameters occurring in the integral conditions. We derive stability conditions for the original three-layer scheme in a specially defined energy norm.     

Finite difference methods for fractional dispersion equations

In this paper, the fractional weighted average finite difference method for space-fractional advection–dispersion equation is proposed, which is based on shifted Grünwald formula. This method is unconditionally stable, consistent and convergent. A numerical example is given, and the numerical results verify the theoretical conclusions.     

Nonlinear explicit difference schemes for solving parabolic equations

A method is proposed for solving initial-boundary-value problems for parabolic equations by means of reducing them to Cauchy problems for systems of ordinary differential equations and applying to the latter nonlinear explicit numerical methods.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 28–30, 1987.     

Convergence of difference schemes for the large-scale ocean dynamics equations

Doklady Mathematics -     

Invariant difference schemes for the Ermakov system

We construct invariant difference schemes for the parametric system of Ermakov equations. By using a difference analog of the Noether theorem, we write out the first three difference integrals of the system. The obtained schemes are integrable exactly to the same extent to which the original differential system is integrable.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

Investigation of difference schemes for quasilinear degenerate parabolic equations

Translated from Issledovaniya po Prikladnoi Matematike, No. 6, pp. 60–73, 1979.     

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.      

Majorizing difference schemes for parabolic equations on unstructured grids

Moscow Physico-Technical Institute. Translated from Matematicheskoe Modelirovanie, Published by Moscow University, Moscow, 1993, pp. 105–113.