  In this paper, we develop an ensemble-based time-stepping algorithm to
efficiently find numerical solutions to a group of linear, second-order
parabolic partial differential equations (PDEs). Particularly, the PDE models
in the group could be subject to different diffusion coefficients, initial
conditions, boundary conditions, and body forces. The proposed algorithm leads
to a single discrete system for the group with multiple right-hand-side vectors
by introducing an ensemble average of the diffusion coefficient functions and
using a new semi-implicit time integration method. The system could be solved
more efficiently than multiple linear systems with a single right-hand-side
vector. We first apply the algorithm to deterministic parabolic PDEs and derive
a rigorous error estimate that shows the scheme is first-order accurate in time
and is optimally accurate in space. We then extend it to find stochastic
solutions of parabolic PDEs with random coefficients and put forth an
ensemble-based Monte Carlo method. The effectiveness of the new approach is
demonstrated through theoretical analysis. Several numerical experiments are
presented to illustrate our theoretical results.
