Convergence Of Finite Difference Scheme Research Articles

On the convergence of finite difference schemes for the heat equation with concentrated capacity

We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is $c\left( x\right) =1+K\delta \left( x-\xi \right). K = \const > 0$ represents the magnitude of the heat capacity concentrated at the point $x=\xi$ . An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained.

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a roughcoefficient function k(x) . We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV , thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

Uniform pointwise convergence of finite difference schemes for quasilinear convection‐diffusion problems

AbstractWe study convergence properties of an upwind difference scheme on layer‐adapted grids for the discretization of a class of singularly perturbed quasilinear two‐point boundary value problems. We derive conditions that are sufficient for uniform convergence in the maximum norm, with respect to the perturbation parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of uniform convergence.

An optimal algorithm for certain boundary value problem

The O( h 4) finite-difference scheme for the second derivative u″( x) leads to a coherent pentadiagonal matrix which is factorized into two tridiagonal matrices. This factorization is used to derive an optimal algorithm for solving a linear system of equations with the pentadiagonal matrix. As an application, a nonlinear system of ordinary differential equations is approximated by an O( h 4) convergent finite-difference scheme. This scheme is solved by the implicit iterative method applying the algorithm at each iteration. A Mathematica module designed for the purpose of testing and using the method is attached.

A Uniformly Convergent Finite Difference Scheme for a Singularly Perturbed Semilinear Equation

Boundary value problems for singularly perturbed semilinear elliptic equations are considered. Special piecewise-uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter. Numerical methods composed of standard monotone finite difference operators and these piecewise-uniform meshes are shown theoretically to be uniformly (with respect to the singular perturbation parameter) convergent. Numerical results are also presented, which indicate that in practice the method is first-order accurate.

Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers

Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers

Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type

Riemann problems with initial data inside elliptic regions are quite different from those for hyperbolic systems. First, we have found that approximate solutions may present persistent oscillations, giving rise to a new type of (measure-valued) waves besides the usual (distributional) ones, shocks and rarefaction waves. Second, any local disturbance of a constant state inside the elliptic region will result in a non-trivial (distributional or, more generally, measure-valued) solution, which is independent of any particular choice of disturbance. For our numerical experiments, we establish two analytical results for testing convergence of finite difference schemes, and for determining expectation values of state functions with respect to the measure-valued solutions when oscillation waves occur. Numerical examples are presented to illustrate those interesting aspects, including the appearance of oscillation waves together with the analysis of the corresponding Young measures.

Convergence acceleration for boundary value problems with singularities using the E-algorithm

In the present work we use the E-algorithm to accelerate the convergence of finite-difference schemes for ordinary differential equations. As a model, a boundary-value problem for a second-order differential equation is considered, where the right-hand side may have different kinds of singularities. An asymptotic error expansion is obtained, enabling the use of the E-algorithm to accelerate the convergence. The applicability and the efficiency of the E-algorithm are discussed and illustrated by numerical examples.

Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

We investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.

On the convergence of finite-difference schemes of high-order accuracy for singular problems

On the convergence of finite-difference schemes of high-order accuracy for singular problems

Finite difference methods for time dependent, linear differential algebraic equations

Recently the authors developed a global reduction procedure for linear, time-dependent DAE that transforms their solutions of smaller systems of ODE's. Here it is shown that this reduction allows for the construction of simple, convergent finite difference schemes for such equations.

Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions: A General Theory

A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

Macrokinetics of laser-induced chemical reactions in gases

This paper deals with the problem of laser radiation interaction with chemical gas. The review of numerical simulation results on the basis of three different models is presented. The main modes of the system behavior and the conditions, with various processes (diffusion, thermoconductivity, thermal diffusion, heat effect of the reaction and others) contributing to their realization, have been established. Theorems about the convergence of finite difference schemes have been proved.

Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach

Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach

Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions: The Corrected Antidiffusive Flux Approach

Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions: The Corrected Antidiffusive Flux Approach

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.

On the convergence of finite-difference schemes for parabolic equations with variable coefficients

We study the convergence of the finite-difference schemes for the first initial-boundary value problem for linear second-order parabolic equations with variable coefficients. Using the bilinear version of the Bramble-Hilbert lemma we obtain estimate of convergence, in discreteW 2 1, 1/2 norm, compatible with the smoothness of generalized solutionu?W 2 ?, ?/2 (Q) (1<??3) and coefficients of equation.

Convergence of Finite Difference Scheme and Analytic Data

Finite difference methods are used widely to approximate solutions of partial differential equations numerically. To build up the difference scheme from the partial differential equation, we should be careful. Formal correspondence of difference schemes to the differential equation is not sufficient for the convergence of the solutions to the exact one. For equations of hyperbolic type, the convergence is assured under the CFL condition [1], which is concerned with the ratio of the space increment to the time one. However, this does not assert that, without CFL condition, every discrete solution diverges. G. Dahlquist [2] exposed an interesting example. He considered the wave equation, and descretized it by the usual finite difference scheme. He showed that if initial data for the wave equation are real analytic, then solutions of difference scheme converge to the solution of the wave equation, even in the absence of the CFL condition. Our results in this issue is in the same direction. Our aim is to show that, in the analytic case, we need not the CFL type condition for the convergence of solutions of the finite difference scheme. We treat of the linear first order system with constant coefficients;

Convergent finite-difference schemes for the equations of the filtration of a fluid with delay

In the paper one constructs convergent difference schemes for the filtration equations of various viscoelastic fluids.

On the rate of convergence of finite difference schemes on nonuniform grids

AbstractFinite difference schemes for some two point boundary value problems are analysed. It is found that for schemes defined on nonuniform grids, the order of the local truncation error does not fully reflect the rate of convergence of the numerical approximation obtained. Numerical results are presented that indicate that this is also the case for higher dimensional problems.