Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation

Abstract

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions, are also presented.

Statistical solutions of the incompressible Euler equations

“…As mentioned before, this algorithm is very similar to the one proposed in Ref. 16 for computing statistical solutions of hyperbolic systems of conservation laws, which in turn was inspired by the ensemble averaging algorithms of Refs. 13 and 25 for computing measure-valued solutions.…”

“…In contrast to Ref. 16 where multi-dimensional hyperbolic systems of conservation laws were considered, we focus on the case of incompressible Euler equations in this paper. Although the concept of statistical solutions is similar in both cases, there are important differences that we highlight in this paper.…”

“…Following Ref. 16, the computation of statistical solutions of (1.1) requires combining the spectral viscosity scheme (4.1) with the following Monte Carlo sampling, • Evolve the samples, using the numerical scheme u Δ i (t) := S Δ tūi , where S Δ t denotes the solution operator, defined by the scheme (4.1). • The approximate statistical solution μ Δ t is given by the so-called empirical measure…”

On convergence of approximate solutions to the compressible Euler system

“…There is a strong piece of evidence, see e.g. Fjordholm et al [24][25][26], that the numerical solutions to the compressible Euler system develop fast oscillations (wiggles) in the asymptotic limit. The resulting object is described by the associated Young measure and it is therefore of interest to know in which sense the limit Euler system is satisfied.…”

On oscillatory solutions to the complete Euler system

“…In this section we consider two classical benchmarks, the Kelvin-Helmholtz and the Richtmyer-Meshkov problems, and illustrate robustness of the concept of K-convergence using, as an example, the FLM (5.3)-(5.5) and GRP (5.16) finite volume schemes. These benchmarks have been also used by Fjordholm et al [31,32], where the weak( * )-convergence of the statistical solutions has been investigated. It is to be pointed out that our theoretical results yield the strong convergence to a dissipative solution, which is the barycenter of the DMV solution.…”