On the Order of Convergence of Solutions of a Difference Equation to a Solution of the Diffusion Equation

Explicit Solutions of the One-Dimensional Heat Equation for a Composite Wall

“…The analysis and test cases used considered constant boundary and initial conditions. Since the stability and convergence depend also on the boundary and initial conditions, as has been pointed out in references [1], [3], and [4], it is quite possible that the introduction of varying conditions would lead to different results as to the usefulness of each method.…”

“…Explicit numerical solutions of the equation of heat conduction in a wall of one material have been widely discussed in the literature. Consideration of the forward difference equation studied in references [2}, [3], [4}, and [6] suggests a variety of ways to handle the solution for a composite wall. This paper is a study of the convergence, stability, comparative accuracy and comparative computing time of three explicit numerical solutions of the heat equation for a wall composed of two materials.…”

“…u(a0, t) = constanti t ^ 0 (3) u(a2, t) = constant2 t ^ 0 and initial conditions: (4) u(x, 0) = constant3 a0 < x < Oi Let each material's thickness, as -a"_i, be divided into N, equal parts of Ax,, and tr into equal parts of At8. Let i denote the subscript associated with the space variable and j the subscript associated with the time variable.…”

On the rate of convergence for parabolic difference schemes, II

“…The solution operators E(t, T ) and Eh(t, T ) clearly fulfill the semigroup property E(t, 7 ) = E(t, T i ) E ( T i 9 T ) > …”

“…We shall now make a short survey of previous results for parabolic problems. Early efforts to deal with problems of this nature for special equations and difference schemes were reported by Juncosa and Young in [7], [ 8 ] , [9] and by Wasow [16]. The more recent work in this area was started by Peetre and Thomte [14].…”