  In this paper, we develop a new reduced basis (RB) method, named as Single
Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations
with homogeneous Dirichlet boundary conditions. The high-fidelity numerical
method adopts the backward Euler scheme and conforming finite elements for the
temporal and spatial discretization, respectively. Under the assumption that
the time step size is sufficiently small and time steps are not very large, we
show that the singular value distribution of the high-fidelity solution matrix
$U$ is close to that of a rank one matrix. We select the eigenfunction
associated with the principal eigenvalue of the matrix $U^\top U$ as the basis
of the Proper Orthogonal Decomposition (POD) method to obtain SEAM and a
parallel SEAM. Numerical experiments confirm the efficiency of the new method.
