Compact difference schemes for heat equation with Neumann boundary conditions

In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949–959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(τ² + h^2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(τ² + h³). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600–616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

New higher-order compact finite difference schemes for 1D heat conduction equations

In this paper, we present two higher-order compact finite difference schemes for solving one-dimensional (1D) heat conduction equations with Dirichlet and Neumann boundary conditions, respectively. In particular, we delicately adjust the location of the interior grid point that is next to the boundary so that the Dirichlet or Neumann boundary condition can be applied directly without discretization, and at the same time, the fifth or sixth-order compact finite difference approximations at the grid point can be obtained. On the other hand, an eighth-order compact finite difference approximation is employed for the spatial derivative at other interior grid points. Combined with the Crank–Nicholson finite difference method and Richardson extrapolation, the overall scheme can be unconditionally stable and provides much more accurate numerical solutions. Numerical errors and convergence rates of these two schemes are tested by two examples.     

Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation

Coupled with the Richardson extrapolation, a new conservative Crank–Nicolson finite difference scheme, which has the accuracy of O(τ²+h⁴)$O({\tau }^2+h^4)$ without refined mesh for the symmetric regularized long wave equation is proposed. The corresponding conservative quantities are discussed, and the existence of numerical solution is proved by the Browder fixed point theorem. The convergence, unconditional stability and uniqueness of the scheme are also derived using the energy method. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm.     

Some results on a heat conduction problem by Myshkis

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u₀>0. The temperature at the origin then decays, and when it reaches u₀, another equal-sized heat impulse is applied at a normalized time τ₁=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ_n, n=2,3,…, when the temperature there has decayed to u₀. This sequence of normalized waiting times τ_n can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91].

A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.     

On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation

We consider an initial boundary‐value problem for the generalized Benjamin–Bona–Mahony equation. A three‐level conservative difference schemes are studied. The obtained algebraic equations are linear with respect to the values of unknown function for each new level. It is proved that the scheme is convergent with the convergence rate of order k – 1, when the exact solution belongs to the Sobolev space of order . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 301–320, 2014     

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

Summary. We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is represents the magnitude of the heat capacity concentrated at the point . 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. Received November 2, 1999 / Revised version received July 24, 2000 / Published online May 4, 2001     

On the convergence of boundary element methods for initial-Neumann problems for the heat equation

In this paper we study boundary element methods for initial-Neumann problems for the heat equation. Error estimates for some fully discrete methods are established. Numerical examples are presented.

     

On a nonlinear generalized max-type difference equation

The boundedness character of positive solutions of the following max-type difference equation

     

Global stability for a class of difference equation

Abstract. In this paper the global stability of difference equation     

Fast high order difference schemes for the time fractional telegraph equation

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ 2-1_σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order and , respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ 2-1_σ method.     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation

In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.