Numerical approximation of statistical solutions of scalar conservation laws

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

Numerical approximation of statistical solutions of scalar conservation laws

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

Uniformly high order accurate essentially non-oscillatory schemes 3

In this paper (a third in a series) the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws are presented. Also presented is a hierarchy of high order accurate schemes which generalizes Godunov's scheme and its second order accurate MUSCL extension to arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and consequently the resulting schemes are highly nonlinear

Twenty-eight years with “Hyperbolic Conservation Laws with Relaxation”

This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the (near) future

Large-time behavior of some numerical schemes: application to the sonic-boom phenomenon

In this thesis we highlight the necessity of employing numerical schemes that preserve the large-time dynamical properties of the continuous system. We focus on Burgers- like equations, which are well known to develop N-waves as intermediate or asymptotic profiles. As we show, not only forward simulations can be distorted; the efficiency of solutions of optimization and inverse design problems might be affected too. In particular, we apply our results to the case of the prediction and control of the sonic- boom phenomenon, modeled by the augmented Burgers equation