We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for a one-dimensional (1D) bipolar Vlasov--Poisson--Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type of decomposition around the local Maxwellian related to the bipolar VPB system in the previous work [H.-L. Li, Y. Wang, T. Yang, and M.-Y. Zhong, <i>Arch. Ration. Mech. Anal.</i>, 228 (2018), pp. 39--127], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to the 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical superposition wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential

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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.

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.

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is to get not only a second order accurate solution but also a second order accurate gradient from each side of the interface. Key to the new method is introducing the jump in the normal derivative of the solution as an augmented variable and rewriting the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivative jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points on or near the interface so that the resulting coefficient matrix is an M-matrix. A multigrid solver is used to solve the linear system of equations, and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface is proved in this paper. Numerical examples for general elliptic interface problems confirm the theoretical analysis and efficiency of the new method.