SIAM Journal on Numerical Analysis
We study stability of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation u'(t) + ∫0 β (t - s)Au(s) ds = 0, t ≥ 0, u(0) = u0, which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space, and β (t) is locally integrable, nonnegative, nonincreasing, convex, and -β'(t) is convex. We establish stability of the method under these hypotheses on β(t). Thus, the method is stable for a wider class of kernel functions β(t) than was previously known. We also extend the class of operators A for which the method is stable.
© 2011 Society for Industrial and Applied Mathematics.
Original Publication Citation
Harris, C. B., & Noren, R. D. (2011). Uniform ι1 behavior of a time discretization method for a volterra integrodifferential equation with convex kernel; stability. SIAM Journal on Numerical Analysis, 49(4), 1553-1571. doi:10.1137/100804656
Harris, Charles B. and Noren, Richard D., "Uniform l1 Behavior of a Time Discretization Method for a Volterra Integrodifferential Equation With Convex Kernel; Stability" (2011). Mathematics & Statistics Faculty Publications. 17.