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

Abstract

Abstract. 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..

Accurate and Efficient Simulations of Hamiltonian Mechanical Systems with Discontinuous Potentials

“…The general case where jumps can be finite, however, appears to be insufficiently studied yet. To that end, this article, along the line of [45,40,48] in which the particle reflection and refraction at the interface are built into the dynamics, proposes four numerical methods, each with distinct applicability. As for general problems, the first method that we recommend to try (among the four plus the penalty method) is the adaptive high-order Integrator 4.…”

“…Due to the non-differentiability of V (•), Hamilton's equation can no longer be used to describe the (meta)particle's global motion. Nevertheless, one can turn to mechanical behavior of particles at the interface to define the solution, as proposed in [45,48,40] (for curved interface in high dimension see [41]): basically, in order for the solution to make sense physically, a corresponding particle should simply evolve locally according to the smooth Hamiltonian dynamics given by Ĥ = 1 2 p T p + U (q), until it hits an interface, and then the particle will either reflect or refract instantaneously, depending on the normal momentum magnitude and whether the jump in V corresponds to a potential barrier or dip across the interface. Then the particle evolves again locally in some D i according to Ĥ, until the next interface hitting.…”

“…q = ∂H/∂p, ṗ = −∂H/∂q) is ill-defined due to indifferentiability of V (q). Following [45,40] (and also its higher-dimensional extension to curved interfaces [41], for Hamiltonian systems with discontinuous Hamiltonians [48], as well as an earlier such approach for well-balanced schemes for the shallow-water equations [70]), we will define a solution based on physical principle, or more precisely, using the classical idea of particle refraction and reflection (used in optics and derived rigorously from Maxwell's equation [39]). See Fig.…”

A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials

“…However, at discontinuity of V, in order to define a physically relevant solution to the initial value problem of (1.10), a particle (or a characteristic defined by (1.10)) is split into two particles with weights corresponding to the transmission and reflection coefficients [11,7]. Since each level set Liouville equation (1.1) is the phase representation of the particle trajectory determined by (1.10), when a particle splits at the interface, one has to add another level set function to describe such a particle or ray splitting.…”

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

“…When hyperbolic equations contain singular coefficients, one usually needs to provide an extra physical condition at the singular points to make the initial or boundary value problems well-posed and to account for the correct physics of waves at the interface or barrier [8,12]. In the case of potential barriers, a natural physical condition is the transmission and reflection conditions, and such conditions can be built into the numerical fluxes in a natural way, in the framework of the Hamiltonian-Preserving schemes [12,13].…”