Computing Multi-valued Physical Observables for the High Frequency Limit of Symmetric Hyperbolic Systems

We develop a level set method for the computation of multi-valued physical observables (density, velocity, energy, etc.) for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions. We take two approaches to derive the method. The first one starts with a decoupled system of an eikonal eikonal for phase S and a transport equations for density ρ: ∂t S+H(x,∇S) = 0, (t,x) ∈ R+×Rn, ∂t ρ+∇x · (ρ∇pH(x,∇xS)) = 0. The main idea is to evolve the density near the n-dimensional bi-characteristic manifold of the eikonal (Hamiltonian-Jacobi) equation, that is identified as the common zeros of n level set functions in phase space. These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously we track a new quantity f by solving again the Liouville equation near the obtained zero level set but with initial density as initial data. The multi-valued density and higher moments are thus resolved by integrating f along the bi-characteristic manifold in the phase directions. The second one uses the high frequency limit of symmetric hyperbolic systems derived by the Wigner transform. This gives rise to Liouville equations in the phase space with measure-valued solution in its initial data. Due to the linearity of the Liouville equation we can decompose the density distribution into products of function, each of which solves the Liouville equation with L∞ initial data on any bounded domain. It yields higher order moments such as energy and energy flux. The main advantages of these new approaches, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: 1) the Liouville equations are solved with L∞ initial data, and a singular integral involving the Dirac-δ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; 2) a local level set method can be utilized to significantly reduce the computation in the phase space. These methods can be used to compute all physical observables for multidimensional problems. Our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix. One such example is the wave equation, which will be studied numerically in this paper.