Convergence Of Difference Scheme Research Articles

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ɛ-uniform maximum norm error estimates O(N−2ln2N(+τ)) and O(N−2(+τ)), respectively, for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, τ is the time meshinterval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition.

Regular and exponential convergence of difference schemes for the heat-conduction equation

The mathematical model for the heat-conduction equation has several special characteristic properties. In this paper, we examine the following property. By increasing time, the solution of the problem tends to the solution of the corresponding elliptic problem. Moreover, the convergence takes place without oscillation and the convergence rate in l 2-norm is the same as the convergence rate of the exponential function to zero. Applying some numerical process, it is not less important to require the preservation of the discrete analogues of the basic qualitative properties of the continuous solution at certain fixed numerical solution (or at all of them). We introduce the (σ, θ)-method which is the generalization both of the well-known Galerkin linear finite element method and the finite difference method and formulate the conditions of the preservation of the regular and exponential convergence. © 1999 Elsevier Science Ltd. All rights reserved.

On the convergence of difference schemes for the diffusion equation of fractional order

For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(h2 + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional order.

Convergence of difference schemes with high resolution for conservation laws

We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions–the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed L p L^{p} estimates of the approximate solutions, H − 1 H^{-1} compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well.

A uniformly convergent difference scheme for the singular perturbation problem of a high order elliptic differential equation

In this paper, we first consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the priori estimation of the solution of the continuous problem. Then, we present an exponential fitted difference scheme and discuss the solution properties of the difference equations. Finally, the uniform convergence of this scheme with respect to the small parameter in the discrete energy norm, is proved.

An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem:\(-\varepsilon^2 u'' + b(x,u) = 0\) , for \(x\in(0,1)\), subject to \(u(0)=u(1)=0\), where \(\varepsilon\in(0,1]\). We assume that \(b_u(x, u)>b_0^2>0\) on\([0, 1] \times {\Bbb R}^1\) , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter\(\varepsilon\) . We present numerical results in support of this result.

An almost fourth order uniformly convergent difference scheme for a semilinear sigularly perturbed reaction-diffusion problem

An almost fourth order uniformly convergent difference scheme for a semilinear sigularly perturbed reaction-diffusion problem

On the convergence of difference schemes for a heat conduction equation

AbstractSufficient conditions are obtained for the convergence of difference schemes for the numerical solution of the Cauchy problem for a heat conduction equation in two space variables. The sufficient conditions are derived in a form similar to those for the convergence of a sequence of linear positive operators in the Korovkin theorem. As an application it is shown that difference schemes that are widely used in practice can easily be checked for convergence by these conditions.

On convergence of difference schemes for nonlinear Schrödinger equations, the Kuramoto-Tsuzuki equation, and reaction-diffusion type systems

On convergence of difference schemes for nonlinear Schrödinger equations, the Kuramoto-Tsuzuki equation, and reaction-diffusion type systems

Convergence of the grid method in the eigenvalue problem for the Helmholtz operator in a right triangle

We prove convergence of difference schemes to generalized solutions of the Helmholtz operator with a coefficient from Lp(Ω), where Ω is a right triangle.

A Comparison of Uniformly Convergent Difference Schemes for Two-Dimensional Convection—Diffusion Problems

Galerkin and Petrov Galerkin finite element methods are used to obtain new finite difference schemes for the solution of linear two-dimensional convection diffusion problems. Numerical estimates are made of the rates of convergence of these schemes, uniformly with respect to the perturbation parameter, and these uniform rates are shown to compare favourably with those of established methods.

Difference scheme for an initial-boundary value problem for linear coefficient-varied parabolic differential equation with a nonsmooth boundary layer function

In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.

Consistency of finite-difference solutions of Einstein's equations.

In the past, arguments have been advanced suggesting that certain finite-difference solutions of the 3+1 form of Einstein's equations suffer from a fundamental inconsistency. Specifically, it has been claimed that freely evolved solutions, where the constraint equations are not explicitly imposed after the initial time, will generally satisfy discrete versions of the constraints to a lower order in the basic mesh spacing h than the truncation order of the discretized evolution equations. This issue is reexamined here, and using the key observation, originally due to Richardson, that a numerical differentiation need not produce an O(${\mathit{h}}^{\mathit{p}\mathrm{\ensuremath{-}}1}$) quantity from an O(${\mathit{h}}^{\mathit{p}}$) one, it is argued that there should be no such inconsistency for convergent difference schemes. Numerical results from a study of spherically symmetric solutions of a massless scalar field minimally coupled to the gravitational field are presented in support of this claim. These results show that the expected convergence of various residual quantities can be achieved in practice.

Difference schemes for fully nonlinear pseudo-hyperbolic systems

The general difference schemes for the first boundary problem of the fully nonlinear pseudohyperbolic systems $$f(x,t,u,u_x ,u_{xx} ,u_t ,u_{tt} ,u_{xt} ,u_{xxt} ) = 0$$ are considered in the rectangular domainQ T ={0≤x≤l, 0≤t≤T}, whereu(x, t) andf(x, t, u, p 1,p 2,r 1,r 2,q 1,q 2) are twom-dimensional vector functions withm ≥ 1 for (x, t) ∈Q T andu, p 1,p 2,r 1,r 2,q 1,q 2 ∈R m . The existence and the estimates of solutions for the finite difference system are established by the fixed point technique. The absolute and relative stability and convergence of difference schemes are justified by means of a series of a prior estimates. In the present study, the existence of unique smooth solution of the original problem is assumed. The similar results for nonlinear and quasilinear pseudo-hyperbolic systems are also obtained.

Convergent difference schemes for the equations of the filtration of a fluid with delay. II

Implicit difference schemes and schemes of variable directions are constructed for the equations of the filtration of fluids with delay of the Maxwell, Oldroyd, and Kelvin-Voight type of order L=1, 2.

Convergence of difference schemes of the reference-operator method to generalized solutions of the axisymmetric Poisson equation

Convergence of difference schemes of the reference-operator method to generalized solutions of the axisymmetric Poisson equation

Convergent difference schemes for equations of motion of Oldroyd fluids

Stable convergent finite-difference schemes are constructed for systems of differential and integrodifferential equations describing the motion of Oldroyd's linear viscoelastic fluids.

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation $$\varepsilon y_{xx} + a(x,t)y_x - b(x,t)y - d(x,t)y_t = f(x,t)$$ on [0, 1]×[0,T], where 0 0,b>0 andd>0. Bounds ony and its derivatives are obtained under certain compatibility assumptions on the data. We examine a family of difference schemes for this problem which is exponentially fitted in thex-variable and uses classical differencing in thet-variable, on rectangular grids which are arbitrarily spaced in both thex andt directions. Under a Courant-Friedrichs-Lewy-type condition, the errors at the grid points are shown to be bounded byC(H+K), whereH(K) is the maximum mesh width in thex(t) direction andC is a constant independent of ɛ,H andK. The corresponding result for a two-point boundary value problem is also derived.

A uniformly convergent second order difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form

In this paper, based on the idea of El-Mistikawy and Werle(1) we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.