This paper discusses three basic issues related to the design of finite difference schemes for unsteadyviscous incompressible flows using vorticity formulations: the boundary condition for vorticity, an efficient time-stepping procedure, and the relation between these schemes and the ones based on velocity-pressure formulation. We show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier. We also show that if we couple a standard centered difference scheme with third- or fourth-order explicit Runge-Kutta methods, the resulting schemes have nocell Reynolds number constraints. For high Reynolds number flows, these schemes are stable under the CFL condition given by the convective terms. Finally, we show that the classical MAC scheme is the same as Thom's formula coupled with second-order centered differences in the interior, in the sense that one can define discrete vorticity in a natural way for the MAC scheme and get the same values as the ones computed from Thom's formula. We use this to derive an efficient fourth-order Runge-Kutta time discretization for the MAC scheme from the one for Thom's formula. We present numerical results for driven cavity flow at high Reynolds number (10 ).

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite-difference or finite-element schemes for the heat equation are stable only if the time step 􏰃t is of the order O(􏰃x2), where 􏰃x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L2-norm of the solution to the integral equation is bounded by cd T d /2 times the norm of the right-hand side. For the Robin problem on the half-space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if 􏰃t < C /κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in the L∞-norm.

Photonic graphene, a photonic crystal with honeycomb structures, has been intensively studied in both theoretical and applied fields. Similar to graphene which admits Dirac Fermions and topological edge states, photonic graphene supports novel and subtle propagating modes (edge modes) of electromagnetic waves. These modes have wide applications in many optical systems. In this paper, we propose a novel gradient recovery method based on Bloch theory for the computation of topological edge modes in the honeycomb structure. Compared to standard finite element methods, this method provides higher order accuracy with the help of gradient recovery technique. This high order accuracy is highly desired for constructing the propagating electromagnetic modes in applications. We analyze the accuracy and prove the superconvergence of this method. Numerical examples are presented to show the efficiency by computing the edge mode for the P-symmetry and C-symmetry breaking cases in honeycomb structures.

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n²) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining