A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation

Abstract

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an α-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by α/(1 − α), independent of the model parameters (the damping coefficient ǫ and the wave velocity γ) and the discretization parameters (the time step τ and the spatial mesh size h). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound α/(1 − α) is very sharp, we find that the occurrence of the damping term −ǫ∆ y t , as well as the large final time T , leads to even faster convergence of the algorithm, especially when α is not very small. We illustrate our theoretical findings with several numerical examples.