A Multilevel Approach to Simulation of a Stochastic Shallow-Water with Rainfall System
Abstract
The one-dimensional shallow water equations (also known as the open channel or one-dimensional Saint-Venant system) is a hyperbolic system of conservation laws that is used to model the overland flow of water in rivers and oceans. Surprisingly, accounting for the additional effect of rainfall on such a system has been much less explored and constitutes the objective of our research.
In [1], with a suitable modification of the boundary conditions we included the effect of rainfall in the d-dimensional shallow water system. The rainfall term was treated here as a deterministic space-time function, but to make any application of this system in forecasting physical situations realistic, the rainfall rate must be modelled as a stochastic term. This stochasticity is modelled by climatologists using so-called weather generators [3], combined with a deterministic component intended to reflect observed or predicted weather patterns. In this work we focus on mathematically more structured stochastic processes such as white noise to provide a proof of concept for our main goal: an efficient numerical method able to compute reliable outputs in reasonable times. Computer modelling of the stochastic rainfall source term via the usual Monte Carlo (MC) method can be expensive even on high performance machines. In this work, we consider a Multilevel Monte Carlo (MLMC) alternative [2], adapting the method developed therein to the specific application of numerical solutions of hyperbolic balance laws. In particular, we consider our solutions across multiple hierarchies of grid size, taking a greater number of samples on the cheaper levels such that the accuracy of our solution remains the same but the computational cost is reduced. We demonstrate the versatility of this method by showing how MLMC can be applied to predict several different variables of interest, and compare the reduction in computational cost against a standard Monte Carlo method. References [1] M. Ersoy, O. Lakkis, and P. Townsend (2017). A Saint-Venant shallow water model for overland flows with precipitation and recharge. Under peer-review as journal article. arXiv, pp. 1-25. url: https://arxiv.org/abs/1705.05470 [2] Giles, Michael B. (2015). Multilevel Monte Carlo methods. Acta Numerica 24, pp. 259-328. [3] IPCC (2013). Stochastic weather generators.- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2018
- Bibcode:
- 2018AGUFMNG21A0802T
- Keywords:
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- 3319 General circulation;
- ATMOSPHERIC PROCESSESDE: 3336 Numerical approximations and analyses;
- ATMOSPHERIC PROCESSESDE: 1990 Uncertainty;
- INFORMATICSDE: 4490 Turbulence;
- NONLINEAR GEOPHYSICS