No abstract uploaded!

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition. We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes.

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $\delta$-singularities. Negative-order norm error estimates for the accuracy of DG approximations to $\delta$-singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $k$-th degree polynomials, at time $t$, the error in the $H^{-(k+2)}$ norm over the whole domain is $(k+1/2)$-th order, and the error in the $H^{-(k+1)}(\mathbb{R}\backslash\mathcal{R}_t)$ norm is $(2k+1)$-th order, where $\mathcal{R}_t$ is the pollution region due to the initial singularity with the width of order $\mathcal{O}(h^{1/2} \log (1/h))$ and $h$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $L^2$ error estimate of $(2k+1)$-th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel's principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $\delta$-singularities.

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently. In this paper, we first discuss an extension of the technique to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one before. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.

The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.