Second-Order Convergence of a Projection Scheme for the Incompressible Navier–Stokes Equations with Boundaries

A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and the pressure is chosen so that the computed velocity satisfies a discrete divergence-free condition. This choice for the pressure and the particular way that the discrete divergence is calculated near the boundary permit the error in the pressure to be controlled and the second-order convergence in the velocity and the pressure to the exact solution to be shown. Some simplifications in the calculation of the pressure in the case without boundaries are also discussed

Stable Fourth Order Stream-Function Methods for Incompressible Flows with Boundaries

Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically

Asymptotic reduction of a porous electrode model for lithium-ion batteries

We present a porous electrode model for lithium-ion batteries using Butler--Volmer reaction kinetics. We model lithium concentration in both the solid and fluid phase along with solid and liquid electric potential. Through asymptotic reduction, we show that the electric potentials are spatially homogeneous which decouples the problem into a series of time-dependent problems. These problems can be solved on three distinguished time scales, an early time scale where capacitance effects in the electrode dominate, a mid-range time scale where a spatial concentration gradient forms in the electrolyte, and a long-time scale where each of the electrodes saturate and deplete with lithium respectively. The solid-phase concentration profiles are linear functions of time and the electrolyte potential is everywhere zero, which allows the model to be reduced to a system of two uncoupled ordinary differential equations. Analytic and numerical results are compared with full numerical simulations and experimental discharge curves demonstrating excellent agreement.Comment: Accepted in SIAM Journal on Applied Mathematic

Discrete Compatibility in Finite Difference Methods for Viscous Incompressible Fluid Flow

Thom's vorticity condition for solving the incompressible Navier-Stokes equations is generally known as a first-order method since the local truncation error for the value of boundary vorticity is first order accurate. In the present paper, it is shown that convergence in the boundary vorticity is actually second order for steady problems and for time-dependent problems when t ? 0. The result is proved by looking carefully at error expansions for the discretization which have been previously used to show second order convergence of interior vorticity. Numerical convergence studies confirm the results. At t = 0 the computed boundary vorticity is first order accurate as predicted by the local truncation error. Using simple model problems for insight we predict that the size of the second order error term in the boundary condition blows up like C= p t as t ! 0. This is confirmed by careful numerical experiments. A similar phenomenon is observed for boundary vorticity computed ..

A Numerical Method for Tracking Curve Networks Moving with Curvature Motion.

A finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions. The prototypical example is that of phase interfaces that meet at prescribed angles, although eutectic junctions and interactions through nonlocal effects are also considered. The method is based on a direct discretization of the underlying parabolic problem and boundary conditions. A linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions. After a singularity in the curve network where the solution is no longer smooth, we demonstrate "almost" second order convergence. A numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions. Finally, different discretizations and methods for implicit time stepping are presented and compared. Key Words: grain growth, interface tracking, finite di..

A numerical framework for singular limits of a class of reaction diffusion problems

We present a numerical framework for solving localized pattern structures of reaction-diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins-Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer-Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries. (C) 2015 Elsevier Inc. All rights reserved