The long time error analysis in the second order difference type method of an evolutionary integral equation with comple
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The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel Da Xu
Received: 23 October 2012 / Accepted: 8 October 2013 © Springer Science+Business Media New York 2013
Abstract We study convergence of the numerical methods in which the second order difference type method is combined with order two convolution quadrature for approximating the integral term of the evolutionary integral equation u (t) +
t
β(t − s) A u (s) ds = 0,
t > 0, u(0) = u0 ,
0
which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space H and β(t) is completely monotonic and locally integrable, but not constant. We establish the convergence properties of the discretization in time in the lt1 (0, ∞; H ) or lt∞ (0, ∞; H ) norm. Keywords Evolutionary integral equation · Completely monotonic kernel · Discretization in time · Second order BDF method · l 1 convergence Mathematics Subject Classifications (2010) 45K05 · 65J08 · 65D32
Communicated by: A. Zhou D. Xu Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People’s Republic of China e-mail: [email protected]
D. Xu
1 Introduction In a series of papers [24–27], we have studied numerical methods for the homogeneous linear equation t ut + β(t − s) A u (s) ds = 0, t > 0, 0
u(0) = u0 ∈ H,
(1.1)
where ut = ∂u/∂t and A is a positive self-adjoint linear operator defined on a dense subspace D (A) of the real Hilbert space H, with a complete eigensystem {λm , ϕm }∞ m=1 . The real-valued kernel β(t) satisfies β ∈ L1 (0, 1), β is completely monotonic on (0, ∞), and 0 ≤ β(∞) < β(0+ ) ≤ ∞.
(1.2)
The present work is a direct continuation of [24] in which we considered the second order accurate scheme for the time discretization of Eq. 1.1 with the completely monotonic kernels Eq. 1.2, and studied the uniform l 1 stability. An earlier paper [26] analyzed the backward Euler method for the time of problem Eq. 1.1, and the uniform l 1 convergence was proved. Achieving the results better than first-order accuracy in time has proven to be challenging, however in some cases extrapolation methods [17, 18] offer a means to improve the accuracy of a low-order method. When the kernel β(t) = t α−1 (0 < α < 1), the numerical solution of problem (1.1) has been extensively studied over the last two decades and a variety of numerical methods have been employed [9–16, 21, 23, 28, 32]. For the second order backward difference time discretization we refer to the works of [9, 10, 13, 14, 21, 23, 28]. For the discontinuous Galerkin in time we refer to the works of [11, 12]. McLean, Sloan and Thomee [16] analyzed an alternative style of time discretization of Eq. 1.1 using the N -point quadrature rule in an approximate Laplace inversion formula, also see McLean, Sloan and Thomee [15] for parabolic integro-differential equations. Recently, Harris and Noren [6] applied the backward Euler scheme and succeeded in showing the uni
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