## A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation

## 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.

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1379639

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Communications in Applied Mathematics and Computational Science

- Additional Journal Information:
- Journal Volume: 12; Journal Issue: 1; Journal ID: ISSN 1559-3940

- Publisher:
- Mathematical Sciences Publishers

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Poisson equation; finite volume methods; high order; embedded boundary

### Citation Formats

```
Devendran, Dharshi, Graves, Daniel, Johansen, Hans, and Ligocki, Terry. A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation. United States: N. p., 2017.
Web. doi:10.2140/camcos.2017.12.51.
```

```
Devendran, Dharshi, Graves, Daniel, Johansen, Hans, & Ligocki, Terry. A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation. United States. doi:10.2140/camcos.2017.12.51.
```

```
Devendran, Dharshi, Graves, Daniel, Johansen, Hans, and Ligocki, Terry. Mon .
"A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation". United States. doi:10.2140/camcos.2017.12.51. https://www.osti.gov/servlets/purl/1379639.
```

```
@article{osti_1379639,
```

title = {A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation},

author = {Devendran, Dharshi and Graves, Daniel and Johansen, Hans and Ligocki, Terry},

abstractNote = {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.},

doi = {10.2140/camcos.2017.12.51},

journal = {Communications in Applied Mathematics and Computational Science},

number = 1,

volume = 12,

place = {United States},

year = {2017},

month = {5}

}

Free Publicly Available Full Text

Publisher's Version of Record

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

##
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

journal, August 1994

- LeVeque, Randall J.; Li, Zhilin
- SIAM Journal on Numerical Analysis, Vol. 31, Issue 4, p. 1019-1044