Uniform \(l^1\) behavior in the Crank-Nicolson methods for a linear Volterra equation with convex kernel
The author studies the time discretization of the Cauchy problem \[ u_t+\int _0^t\beta(t-s)Lu(s)ds=0,\, t>0, \,u(0)=u_0, \] where \(L\) is a self-adjoint densely defined linear operator on the Hilbert space having a complete eigensystem \({\{\lambda_m,\varphi_m\}}_{m=1}^\infty\). The kernel \(\beta\in C(0,\infty)\cap L^1(0,1)\) is real-valued, nonnegative, nonincreasing and convex, \(-\beta '\) is also convex. The discretization in time is based on using the Crank-Nicolson method and on the trapezoidal rule. The integral term is also approximated by the trapezoidal rule. Uniform error estimates are also studied and some numerical examples are given.
linear Volterra integro-differential equation
scientific article; zbMATH DE number 6418667
scientific article