Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
Journal of Computational Physics
Journal of Computational Physics
Multi-phase computations in geometrical optics
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
Journal of Computational Physics
An Eulerian method for capturing caustics
Journal of Computational Physics
High-frequency wave propagation by the segment projection method
Journal of Computational Physics
Using K-branch entropy solutions for multivalued geometric optics computations
Journal of Computational Physics
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
Mathematics of Computation
Local level set method in high dimension and codimension
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
Journal of Computational Physics
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Journal of Computational Physics
Computing multi-valued velocity and electric fields for 1D Euler--Poisson equations
Applied Numerical Mathematics
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
A quadrature-based third-order moment method for dilute gas-particle flows
Journal of Computational Physics
Journal of Computational Physics
Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics
Journal of Scientific Computing
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
Journal of Computational Physics
High order numerical methods to two dimensional delta function integrals in level set methods
Journal of Computational Physics
A level set approach for dilute non-collisional fluid-particle flows
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
Numerical Approximation of the Euler-Poisson-Boltzmann Model in the Quasineutral Limit
Journal of Scientific Computing
Hi-index | 31.49 |
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 weakly coupled system of an eikonal equation for phase S and a transport equation for density @r:@?"tS+H(x,@?S)=0,(t,x)@?R^+xR^n,@?"t@r+@?"x.(@r@?"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, which is identified as the common zeros of n level set functions in phase space (x,k)@?R^2^n. 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=@r(t,x,k)|det(@?"k@f)| by solving again the Liouville equation near the obtained zero level set @f=0 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-@d 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 multi-dimensional 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.