Vol. 12, No. 1, 2017

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
A fourth-order Cartesian grid embedded boundary method for Poisson's equation

Dharshi Devendran, Daniel T. Graves, Hans Johansen and Terry Ligocki

Vol. 12 (2017), No. 1, 51–79
Abstract

In this paper, we present a fourth-order algorithm to solve Poisson’s equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

Keywords
Poisson equation, finite volume methods, high order, embedded boundary
Mathematical Subject Classification 2010
Primary: 65M08, 65M50
Milestones
Received: 25 March 2016
Revised: 19 December 2016
Accepted: 30 January 2017
Published: 8 May 2017
Authors
Dharshi Devendran
Applied Numerical Algorithms Group (ANAG)
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States
Daniel T. Graves
Computational Research Division
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States
Hans Johansen
Applied Numerical Algorithms Group (ANAG)
Computational Research Division
Lawrence Berkeley National Laboratory
MS 50A1148
One Cyclotron Road
Berkeley, CA 94720
United States
Terry Ligocki
Applied Numerical Algorithms Group (ANAG)
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States