A variational Crank–Nicolson ensemble Monte Carlo algorithm for a heat equation under uncertainty

Abstract

In this paper, a new fully discrete variational Crank–Nicolson ensemble (FVE/CN) scheme is established for the heat equation with uncertainty, then a fully discrete variational Crank–Nicolson ensemble Monte Carlo algorithm (FVEMC/CN) is given. Using the FVEMC/CN algorithm, one just needs to solve a single linear system with multiple right-hand side vectors in a group at each time step. The stability and the error estimates are presented to indicate the proposed algorithm is second-order convergent regarding the time. Finally, numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed approaches.