Numerical approximation of statistical solutions of scalar conservation laws
For researchers in computational fluid dynamics and uncertainty quantification, this work provides provably convergent numerical methods for statistical solutions of conservation laws, with demonstrated efficiency improvements.
The paper proposes efficient numerical algorithms combining finite volume methods with Monte Carlo and multi-level Monte Carlo for approximating statistical solutions of scalar conservation laws, proving convergence and showing that multi-level Monte Carlo offers considerable efficiency gains over standard Monte Carlo.
We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multi-level Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multi-point statistical quantities of interest are also presented.