A Stochastic Galerkin Method for Hamilton--Jacobi Equations with Uncertainty

Abstract

We develop a class of stochastic numerical schemes for Hamilton--Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the Hamilton--Jacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the Jin--Xin relaxation approximation in physical space for shock capturing. We provide an error estimate for the gPC stochastic Galerkin approximation to smooth solutions, and show that our numerical formulation preserves the symmetry and hyperbolicity of the underlying system, which allows one to efficiently quantify the uncertainty of the Hamilton--Jacobi equations due to random inputs, as demonstrated by the numerical examples.