Numerical Efficiency in Monte Carlo Simulations—Case Study of a River Thermodynamic Model

Trade-offs between precision of numerical solutions to deterministic models of the environment, and the number of model realizations achievable within a framework of Monte Carlo simulation, are investigated and discussed. A case study of a model of river thermodynamics is employed. It is shown that the tractability of Monte Carlo simulation relies on adaptation of the numerical solution time-step, giving results with a guaranteed error in the time domain as well as near-optimum speed of calibration under any chosen accuracy criteria. Time-step control is implemented using two adaptive Runge–Kutta methods: a second order scheme with first order error estimator, and an embedded fourth-fifth order scheme. In the case study, where the effects of sparse and imprecise data dominate the overall modeling error, both the schemes appear adequate. However, the higher order scheme is concluded to be generally more reliable and efficient, and has wide potential to improve the value of applying the Monte Carlo method to environmental simulation. The problem of reconciling spatial error with the specified temporal error is discussed.