Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant

The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition-has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce two ideas to remedy this: 1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant. We show that this is many times faster than the usual iterative minimization of a singular value. 2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non-simply-connected domains. We implement the new method in two dimensions using spectrally accurate Nystrom product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including ones with strong exterior resonances.Comment: 21 pages, 7 figures, submitted to SIAM Journal of Numerical Analysis. Updated a duplicated picture. All results unchange

Evolution of dark energy reconstructed from the latest observations

We reconstruct evolution of the dark energy (DE) density using a nonparametric Bayesian approach from a combination of latest observational data. We caution against parameterizing DE in terms of its equation of state as it can be singular in modified gravity models, and using it introduces a bias preventing negative effective DE densities. We find a $3.7\sigma$ preference for an evolving effective DE density with interesting features. For example, it oscillates around the $\Lambda$CDM prediction at $z\lesssim0.7$, and could be negative at $z\gtrsim2.3$; dark energy can be pressure-less at multiple redshifts, and a short period of cosmic deceleration is allowed at $0.1 \lesssim z\lesssim 0.2$. We perform the reconstruction for several choices of the prior, as well as a evidence-weighted reconstruction. We find that some of the dynamical features, such as the oscillatory behaviour of the DE density, are supported by the Bayesian evidence, which is a first detection of a dynamical DE with a positive Bayesian evidence. The evidence-weighted reconstruction prefers a dynamical DE at a $(2.5\pm0.06)\sigma$ significance level.Comment: 10 pages, 6 figures, 2 tables; accepted for publication in ApJ Letter

Coupled Continuum and Molecular Model of Flow through Fibrous Filter

A coupled approach combining the continuum boundary singularity method (BSM) and the molecular direct simulation Monte Carlo (DSMC) is developed and validated using Taylor-Couette flow and the flow about a single fiber confined between two parallel walls. In the proposed approach, the DSMC is applied to an annular region enclosing the fiber and the BSM is employed in the entire flow domain. The parameters used in the DSMC and the coupling procedure, such as the number of simulated particles, the cell size, and the size of the coupling zone are determined by inspecting the accuracy of pressure drop obtained for the range of Knudsen numbers between zero and unity. The developed approach is used to study flowfield of fibrous filtration flows. It is observed that in the partial-slip flow regime, Kn ⩽ 0.25, the results obtained by the proposed coupled BSM-DSMC method match the solution by BSM combined with the heuristic partial-slip boundary conditions. For transition molecular-to-continuum Knudsen numbers, 0.25 \u3c Kn ⩽ 1, the difference in pressure drop and velocity between these two approaches is significant. This difference increases with the Knudsen number that confirms the usefulness of coupled continuum and molecular methods in numerical modeling of transition low Reynolds number flows in fibrous filters