Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws

We study two intrusive methods for uncertainty propagation in scalar conservation laws based on their kinetic formulations. The first method uses convolutions with Jackson kernels based on expansions on an orthogonal family of polynomials, and we prove that it satisfies bounded variations and converges to the entropy solution but with a spurious damping phenomenon. Therefore we introduce a second method, which is based on projection on layered Maxellians and which arises as a minimization of entropy. Our construction of layered Maxwellians relies on the Bojanic--Devore theorem about best $L^1$ polynomial approximation. This new method, denoted below as a kinetic polynomial method, satisfies the maximum principle by construction as well as partial entropy inequalities and thus provides an alternative to the standard method of moments which, in general, does not satisfy the maximum principle. Simple numerical simulations for the Burgers equation illustrate these theoretical results.