Uniform $$l^{1}$$ behavior in the Crank---Nicolson methods for a linear Volterra equation with convex kernel

• Authors: Da Xu
• Venue: Calcolo: a quarterly on numerical analysis and theory of computation
• Year: 2014

We study the time discretization of the Cauchy problem $$\begin{aligned} u_{t}+\int _{0}^{t}\,\beta (t-s)\,L\,u\,(s)\;ds = 0,\quad t0, \quad u(0)=u_{0}, \end{aligned}$$where $$L$$ is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigen system $$\{\lambda _{m},\; \varphi _{m}\}_{m=1}^{\infty }$$, and the subscript denotes differentiation with respect to $$t$$. The real valued kernel $$\beta \in \,C(0,\,\infty )\bigcap \,L^{1}(0,\,1)$$ is assumed to be nonnegative, nonincreasing and convex, and $$-\beta ^{\prime }$$ is convex. The equation is discretized in time by Crank---Nicolson method based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature rule, constructed again from the trapezoidal rule, is used to approximate the integral term. The results and methods extend and simulate numerically those introduced by Carr and Hannsgen (SIAM J Math Anal 10:961---984, 1979) and (SIAM J Math Anal 13:459---483, 1982) for integrability with respect to continuous solutions. The uniform error estimates of the discretization in time are derived in the $$ l^{\infty }_{t}(0,\,\infty ;H) $$ norm. Some simple numerical examples illustrate our theoretical error bounds.