An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs
For computational scientists solving multiple parabolic PDEs, this algorithm offers efficiency gains through ensemble averaging, though it is an incremental improvement over existing ensemble methods.
The paper develops an ensemble-based time-stepping algorithm for solving groups of linear parabolic PDEs with varying coefficients, achieving first-order temporal and optimal spatial accuracy. The method reduces computational cost by solving a single discrete system with multiple right-hand sides, and is extended to stochastic PDEs via ensemble Monte Carlo.
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.