  In this paper, we propose a direct parallel-in-time (PinT) algorithm for
time-dependent problems with first- or 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 $\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$. 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.
