# High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

## Abstract

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materialsmore »

- Authors:

- Rensselaer Polytechnic Inst., Troy, NY (United States). Dept. of Mathematical Sciences

- Publication Date:

- Research Org.:
- Rensselaer Polytechnic Inst., Troy, NY (United States)

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

- OSTI Identifier:
- 1393833

- Alternate Identifier(s):
- OSTI ID: 1549294

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 352; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; wave equations; electromagnetics; upwind methods; overlapping grids

### Citation Formats

```
Angel, Jordan B., Banks, Jeffrey W., and Henshaw, William D. High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.09.037.
```

```
Angel, Jordan B., Banks, Jeffrey W., & Henshaw, William D. High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form. United States. doi:10.1016/j.jcp.2017.09.037.
```

```
Angel, Jordan B., Banks, Jeffrey W., and Henshaw, William D. Thu .
"High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form". United States. doi:10.1016/j.jcp.2017.09.037. https://www.osti.gov/servlets/purl/1393833.
```

```
@article{osti_1393833,
```

title = {High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form},

author = {Angel, Jordan B. and Banks, Jeffrey W. and Henshaw, William D.},

abstractNote = {High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. Finally, the upwind scheme is shown to be robust and provide high-order accuracy.},

doi = {10.1016/j.jcp.2017.09.037},

journal = {Journal of Computational Physics},

number = ,

volume = 352,

place = {United States},

year = {2017},

month = {9}

}

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