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# Methods and Applications of Analysis

## Volume 20 (2013)

### Number 4

### Special issue dedicated to the 70th birthday of Stanley Osher: Part I

Guest Editor: Chi-Wang Shu, Brown University

### A virtual node algorithm for Hodge decompositions of inviscid flow problems with irregular domains

Pages: 439 – 455

DOI: http://dx.doi.org/10.4310/MAA.2013.v20.n4.a8

#### Authors

#### Abstract

We present an efficient discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions using a virtual node framework. The method is designed for use in the exact projection discretization of incompressible flow. We leverage the Poisson framework initially developed in [1] and [2]. This approach uses a signed distance function to represent the irregular domain embedded in a Cartesian grid and uses a variational approach to create a symmetric positive definite linear system. We present a novel modification to the previous approach that yields a 5-point stencil (7-point in 3D) across the entire computational domain. The original algorithm required a 9-point stencil (27-point in 3D) near the embedded irregular boundary. We show that this new condensed stencil enables a decomposition of the form $\mathbf{A} = \mathbf{G}^T \mathbf{M}^{-1} \mathbf{G}$, where $\mathbf{M}$ is a diagonal weighting matrix and $\mathbf{G}$ and $\mathbf{D} = -{\mathbf{G}^T}$ are diagonal scalings of the standard central-difference gradient and divergence operators. We use this factored form as the basis of our discrete Hodge decomposition and show that this can be readily used for exact projection in incompressible flow. Numerical experiments suggest our method is second-order in $L^{\infty}$ for pressures and first-order in $L^{\infty}$, second-order in $L^1$ for velocities.

#### Keywords

level sets, Hodge decomposition, incompressible flow, MAC grid, virtual node algorithms

#### 2010 Mathematics Subject Classification

76Bxx, 76M10, 76M20