A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations

In this paper, we propose a direct parallel-in-time (PinT) algorithm for timedependent problems with firstor second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix V grows as n is increased, where n is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing V and V −1, by which we prove that Cond2(V ) = O(n2). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.