Uniform Error Estimates for Navier--Stokes Flow with an Exact Moving Boundary Using the Immersed Interface Method

Abstract

We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Difference operators are corrected to O(h) near the interface using the immersed interface method, adding terms related to the jumps, on a regular grid with spacing h and periodic boundary conditions. The force at the interface is assumed known within an error tolerance; errors in the interface location are not taken into account. The error in velocity is shown to be uniformly O(h 2 | log h| 2 ), even at grid points near the interface, and, up to a constant, the pressure has error O(h 2 | log h| 3 ). The proof uses estimates for finite difference versions of Poisson and diffusion equations which exhibit a gain in regularity in maximum norm.

Well‐Posedness and Global Behavior of the Peskin Problem of an Immersed Elastic Filament in Stokes Flow

“…In the case of D 1 2 , we use the second statements in Propositions 3.8, 3.9, and 3.12 with the choice of˛D 3 2 and the result follows from the same arguments. In the case of D 1 2 , we use the second statements in Propositions 3.8, 3.9, and 3.12 with the choice of˛D 3 2 and the result follows from the same arguments.…”

“…Taking T D minfT 1 ; T 2 g gives the desired result. In the case of D 1 2 , we use the second statements in Propositions 3.8, 3.9, and 3.12 with the choice of˛D 3 2 and the result follows from the same arguments. Proposition 3.13 gives the local existence of a solution to the Peskin problem for any initial data X 0 2 h 1; .S 1 / with jX 0 j > 0.…”

“…From a numerical standpoint, unless some smoothness is established, it may not be clear whether the various methods are approximating the same solution. The wealth of numerical methods that can be used to tackle this problem has the potential to make the Peskin problem a standard testbed for the numerical analysis of FSI problems [3,28].…”

An Eulerian formulation of immersed interface method for membranes with tangential stretching

“…In [25], an immersed interface method with direct discretization of the Navier-Stokes equations was proposed. In [1], Beale considered an immersed interface method with a direct finite difference on a periodic domains. The convergence of the immersed interface method has been shown to be nearly second order with the assumption that the forces are given exactly.…”

Sharp error estimates of a fourth‐order compact scheme for a Poisson interface problem

“…The aim of this paper is to establish the sharp error analysis of the CFDS for the simple problem (1)–(4). Following the strategy in [1, 2], the local truncation errors at irregular grid nodes with O ( h 3 ) accuracy can be written as the discrete divergence of some grid functions, which is smaller than the lower‐order local truncation errors by a factor of h . In terms of the grid functions, we first show that the CFDS can yield fourth‐order accuracy in the discrete ℓ 2 ‐norm for the approximate solution and its gradient.…”