Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond

This paper reviews some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coefficients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuous bottom topography and the Hamiltonian preserving schemes for Liouville equations with discontinuous Hamiltonians is illustrated. Various developments of numerical methods with applications in high frequency waves through interfaces, and in multiscale coupling between classical and quantum mechanics, are discussed.