We present numerical schemes for the incompressible Navier–Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematic equations. The result is a class of extremely efficient Navier–Stokes solvers. Full time accuracy is achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we prove unconditional stability of the new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier–Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C0 elements to implement.

Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of nonlinearity
here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the band edge is lower than the limit power value on the band edge. Through numerical computations of the power curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical eigenvalue formula and numerically computed eigenvalues shows excellent agreement.

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.

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson–Nernst–Planck (PNP) equations, which have found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR\&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR\&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR\&WENO is accurate and have a high resolution. Moreover, we evaluate and compare the performance between the uniform mesh WENO scheme and the parallel AMR\&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR\&WENO method.