Moysey Brio

Mach reflection for the two-dimensional Burgers equation

Abstract We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations.

Weak shock reflection

An asymptotic analysis of the regular and Mach reflection of weak shocks leads to shock reflection problems for the unsteady transonic small disturbance equation. Numerical solutions of this equation resolve the von Neumann triple point paradox for weak shock Mach reflection. Related equations describe steady transonic shock reflections, weak shock focusing, and nonlinear hyperbolic waves at caustics.

The von Neumann paradox in weak shock reflection

We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.

Multimode interference in circular step-index fibers studied with the mode expansion approach

The mode expansion approach in vectorial form, using a complete set of guided modes of a circular step-index fiber (SIF), is developed and applied to analyze multimode interference in multimode fibers (MMFs) for the first time, to the best of our knowledge. The complete set of guided modes of an SIF is defined based on its modal properties, and a suitable modal orthogonality relation is identified to evaluate the coefficients in a mode expansion. An algorithm, adaptive to incident fields, is then developed to systematically and efficiently perform mode expansion in highly MMFs. The mode expansion approach is successfully applied to investigate the mode-selection properties of coreless fiber segments incorporated in multicore fiber lasers and the self-imaging in MMFs.

Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model

In this paper we present a numerical method for solving a three-dimensional cold-plasma system that describes electron gas dynamics driven by an external electromagnetic wave excitation. The nonlinear Drude dispersion model is derived from the cold-plasma fluid equations and is coupled to the Maxwells field equations. The Finite-Difference Time-Domain (FDTD) method is applied for solving the Maxwells equations in conjunction with the time-split semi-implicit numerical method for the nonlinear dispersion and a physics based treatment of the discontinuity of the electric field component normal to the dielectric-metal interface. The application of the proposed algorithm is illustrated by modeling light pulse propagation and second-harmonic generation (SHG) in metallic metamaterials (MMs), showing good agreement between computed and published experimental results.

Second-order accurate FDTD space and time grid refinement method in three space dimensions

We present an algorithm based on the finite-difference time-domain method for local refinement of a three-dimensional computational grid in space and time. The method has second-order accuracy in space and time as verified in the numerical examples. A number of test cases with material traverse normal to the grid interfaces were used to assess the long integration time stability of the algorithm. Resulting improvements in the computation time are discussed for a photonic crystal microcavity design that exhibits a sensitive dependence of the quality factor on subwavelength geometrical features

Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks

Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfven waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfven waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B · ∇ × B ≠0 and/or B · ∇ × u ≠0, where B and u are the magnetic field induction and fluid velocity respectively).

Nonlinear and three-wave resonant interactions in magnetohydrodynamics

Hamiltonian and variational formulations of equations describing weakly nonlinear magnetohydrodynamic (MHD) wave interactions in one Cartesian space dimension are discussed. For wave propagation in uniform media, the wave interactions of interest consist of (a) three-wave resonant interactions in which highfrequency waves may evolve on long space and time scales if the wave phases satisfy the resonance conditions: (b) Burgers self-wave steepening for the magnetoacoustic waves, and (c) mean wave field effects, in which a particular wave interacts with the mean wave field of the other waves. The equations describe four types of resonant triads: slow-fast magnetoacoustic wave interaction, Alfven-entropy wave interaction, Alfven-magnetoacoustic wave interaction, and magnetoacoustic-entropy wave interaction. The formalism is restricted to coherent wave interactions. The equations are used to investigate the Alfven-wave decay instability in which a large-amplitude forward propagating Alfven wave decays owing to three-wave resonant interaction with a backward-propagating Alfven wave and a forward-propagating slow magnetoacoustic wave. Exact solutions of the equations for Alfven-entropy wave interactions are also discussed.

Application of Post's formula to optical pulse propagation in dispersive media

In this paper we consider the propagation of optical pulses in dielectric media with nontrivial dispersion relations. In particular, we implement Posts Laplace transform formula to invert in time the Fourier-Laplace space coefficients which arise from the joint space solution of the optical dispersive wave equation. Due to the inefficiency of a direct application of this formula, we have considered and present here two more efficient implementations. In the first, the Gaver-Post method, we utilize the well known Gaver functionals but store intermediate calculations to improve efficiency. The second, the Bell-Post method, involves an analytic reformulation of Posts formula such that knowledge of the dispersion relation and its derivatives are sufficient to invert the coefficients from Laplace space to time. Unlike other approaches to the dispersive wave equation which utilize a Debye-Lorentzian assumption for the dispersion relation, our algorithm is applicable to general Maxwell-Hopkinson dielectrics. Moreover, we formulate the approach in terms of the Fourier-Laplace coefficients which are characteristic of a dispersive medium but are independent of the initial pulse profile. They thus can be precomputed and utilized when necessary in a real-time system. Finally, we present an illustration of the method applied to optical pulse propagation in a free space and in two materials with Cole-type dispersion relations, room temperature ionic liquid (RTIL) hexafluorophosphate and brain white matter. From an analysis of these examples, we find that both methods perform better than a standard Post-formula implementation. The Bell-Post method is the more robust of the two, while the Gaver-Post is more efficient at high precision and Post formula approximation orders.

Overlapping Yee FDTD Method on Nonorthogonal Grids

We propose a new overlapping Yee (OY) method for solving time-domain Maxwell’s equations on nonorthogonal grids. The proposed method is a direct extension of the Finite-Difference Time-Domain (FDTD) method to irregular grids. The OY algorithm is stable and maintains second-order accuracy of the original FDTD method, and it overcomes the late-time instability of the previous FDTD algorithms on nonorthogonal grids. Numerical examples are presented to illustrate the accuracy, stability, convergence and efficiency of the OY method.