We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter ε∈(0,1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. 0<ε≪1, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with O(ε)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed O(1/ε). This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h^2+τ^2/ε) and O(h^2+τ+ε) with h mesh size and τ time step. Thus we obtain a uniform error bound at O(h^2+τ) for 0<ε≤1. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and ε-dependent error bounds between the solutions of KGZ and its limiting model when ε→0+. Finally, numerical results are reported to confirm our error bounds.

The Euler equation of compressible flows is solved by the finite volume method, where high order accuracy is achieved by the reconstruction of each component of upwind fluxes of a flux splitting using the biased averaging procedure. Compared to the solution reconstruction in Godunov-type methods, its implementation is simple and easy, and the computational complexity is relatively low. This approach is parameter-free and requires neither a Riemann solver nor field-by-field decomposition. The numerical results from both dynamic and steady state calculations demonstrate the accuracy and robustness of this approach. Some techniques for the acceleration of the convergence to the steady state are discussed, including multigrid and multistage Runge–Kutta time methods.

In this paper, we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition, we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the nonzero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, traditional numerical methods typically fail since the solutions of the advection-diffusion equations develop sharp gradients. Instead of solving the Fokker--Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We propose an effective numerical integrator based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem

Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with non-triangular faces as holes, by either using the nodes' location or using a landmark-based scheme without node location. The conformal map is computed with Ricci flow such that all the non-triangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossip-style computation. The method applies to both unit disk graph models and quasi-unit disk graph

Shape optimization aims to optimize an objective function by changing the shape of the computational domain. In recent years, shape optimization has received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis of the problem; see e.g., [6] and references therein.