Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

We present a simple but effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer poten- tials in two and three spatial dimensions. The method relies on the use of Green’s third identity and local Taylor-like interpolations of density functions […]

Equation Level Matching: An Extension of the Method of Matched Asymptotic Expansion for Problems of Wave Propagation

We introduce an alternative to the method of matched asymptotic expansions. In the “traditional” implementation, approximate solutions, valid in different (but overlapping) regions, are matched by using “intermediate” variables. Here we propose to match at the level of the equations involved, via a “uniform expansion” whose equations enfold those of […]

Non-specular reflection of walking droplets

Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale […]

Study of a model equation in detonation theory

We analyze properties of a nonlocal hyperbolic balance law equation that we previously proposed to model the dynamics of unstable detonation waves. We show that much of the complexity present in one-dimensional detonations can be captured by a properly forced Burgers’ equation. Furthermore we employ a combination of numerical and analytical tools to investigate the […]

Model for shock wave chaos

We propose a simple model equation that predicts chaotic shock waves, similar to those in detonations in chemically reacting mixtures. The equation is given on the half line, with the shock located at x=0.  This equation retains the essential physics needed to reproduce many properties of detonations in gaseous reactive mixtures: […]