  We propose a numerical method to solve parameter-dependent hyperbolic partial
differential equations (PDEs) with a moment approach, based on a previous work
from Marx et al. (2020). This approach relies on a very weak notion of solution
of nonlinear equations, namely parametric entropy measure-valued (MV)
solutions, satisfying linear equations in the space of Borel measures. The
infinite-dimensional linear problem is approximated by a hierarchy of convex,
finite-dimensional, semidefinite programming problems, called Lasserre's
hierarchy. This gives us a sequence of approximations of the moments of the
occupation measure associated with the parametric entropy MV solution, which is
proved to converge. In the end, several post-treatments can be performed from
this approximate moments sequence. In particular, the graph of the solution can
be reconstructed from an optimization of the Christoffel-Darboux kernel
associated with the approximate measure, that is a powerful approximation tool
able to capture a large class of irregular functions. Also, for uncertainty
quantification problems, several quantities of interest can be estimated,
sometimes directly such as the expectation of smooth functionals of the
solutions. The performance of our approach is evaluated through numerical
experiments on the inviscid Burgers equation with parametrised initial
conditions or parametrised flux function.
