The numerical analysis of the weighted average method for heat conduction equation in one dimension

Abstract

In this paper, we will consider the weighted average method for heat conduction equation in one dimension. When θ � 1 the truncation error of the scheme is O(τ + h 2 ) ;i fθ = 1 the truncation error of the scheme is O(τ 2 + h 2 ) .W hen 0 θ 1 , the scheme is stable if and only ifγ 1 (1 − 2θ) �1 and when 1 θ 1, the scheme is stable for all γ .T he numerical experiments are illustrated at the last. u ∂x2 , ( a> 0), −∞ <x< +∞, 0 t T u(x, 0) = ϕ(x), −∞ <x< +∞ (1) For this model problem an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle, or by Fourier analysis (5). Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are as- sociated with explicit-implicit splitting of the problem operator. These schemes belong to the class of uncondition ally stable schemes, but they demonstrate bad approximation properties. In (8) P.N. Vabi shchevich proposes a multilevel modification of the alternating triangle method, which demonstrates better property -es in terms of accuracy. In (6) Rooholah Abedian proposes a new WENO finite differ- enceprocedurefor nonlineardegenerateparabolicequationswhich may containdiscontinuoussolutions. Their scheme is based on the method of lines, with a high-order accurate conservative approximation to each of the diffuse -on terms. In (7) Allaberen Ashyralyev considers the r-modified Crank-Nicholson difference schemes for the approximate solution of the ultra-parabolic equations. Explicit scheme is simple but conditionally stable. Implicit methods are stable, however they can take much longer to compute than explicit methods. So we consider to built the weighted average scheme to combine the advantages of both type scheme. In the present paper, we are going to analyze the truncation error and stability of the weighted average scheme in theory or in numeration. The paper is organized as following: The numerical schemes for heat conduction equations in one dimension; numerical experiments; summary.