Numerical analysis of uncertain temperature field by stochastic finite difference method

Based on the combination of stochastic mathematics and conventional finite difference method, a new numerical computing technique named stochastic finite difference for solving heat conduction problems with random physical parameters, initial and boundary conditions is discussed. Begin with the analysis of steady-state heat conduction problems, difference discrete equations with random parameters are established, and then the computing formulas for the mean value and variance of temperature field are derived by the second-order stochastic parameter perturbation method. Subsequently, the proposed random model and method are extended to the field of transient heat conduction and the new analysis theory of stability applicable to stochastic difference schemes is developed. The layer-by-layer recursive equations for the first two probabilistic moments of the transient temperature field at different time points are quickly obtained and easily solved by programming. Finally, by comparing the results with traditional Monte Carlo simulation, two numerical examples are given to demonstrate the feasibility and effectiveness of the presented method for solving both steady-state and transient heat conduction problems.