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Morse and Ingard give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed earlier for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner.

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nystr\"om method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.

We report the experimental determination of the structure and response to applied electric field of the lower-temperature nematic phase of the previously reported calamitic compound 4-[(4-nitrophenoxy)carbonyl]phenyl2,4-dimethoxybenzoate (RM734). We exploit its electro-optics to visualize the appearance, in the absence of applied field, of a permanent electric polarization density, manifested as a spontaneously broken symmetry in distinct domains of opposite polar orientation. Polarization reversal is mediated by field-induced domain wall movement, making this phase ferroelectric, a 3D uniaxial nematic having a spontaneous, reorientable, polarization locally parallel to the director. This polarization density saturates at a low temperature value of ~ 6 microcoulombs/cm-sqd, the largest ever measured for an organic material or for any fluid. This polarization is comparable to that of solid state ferroelectrics, and is close to the average value obtained by assuming perfect, polar alignment of molecular long axes in the nematic. We find a host of spectacular optical and hydrodynamic effects driven by ultra-low applied field (E~1V/cm), produced by the coupling of the large polarization to nematic birefringence and flow. Electrostatic self-interaction of the polarization charge renders the transition from the nematic phase mean-field-like and weakly first-order, and controls the director field structure of the ferroelectric phase. Atomistic molecular dynamics simulation reveals short-range polar molecular interactions that favor ferroelectric ordering, including a tendency for head-to-tail association into polar, chain-like assemblies having polar lateral correlations. These results indicate a significant potential for transformative new nematic science and technology based on the enhanced understanding, development, and exploitation of molecular electrostatic interaction., Comment: main paper: 23 pages; SI 34 pages

We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence with adaptive refinement to manage error from function extension.

We proposed an efficient iterative thresholding method for multi-phase image segmentation. The algorithm is based on minimizing piecewise constant Mumford-Shah functional in which the contour length (or perimeter) is approximated by a non-local multi-phase energy. The minimization problem is solved by an iterative method. Each iteration consists of computing simple convolutions followed by a thresholding step. The algorithm is easy to implement and has the optimal complexity $O(N \log N)$ per iteration. We also show that the iterative algorithm has the total energy decaying property. We present some numerical results to show the efficiency of our method., Comment: 14 pages, 21 figures