# Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement

## Abstract

This paper describes an approach for the numerical solution of time-dependent partial differential equations in complex three-dimensional domains. The domains are represented by overlapping structured grids, and block-structured adaptive mesh refinement (AMR) is employed to locally increase the grid resolution. In addition, the numerical method is implemented on parallel distributed-memory computers using a domain-decomposition approach. The implementation is flexible so that each base grid within the overlapping grid structure and its associated refinement grids can be independently partitioned over a chosen set of processors. A modified bin-packing algorithm is used to specify the partition for each grid so that the computational work is evenly distributed amongst the processors. All components of the AMR algorithm such as error estimation, regridding, and interpolation are performed in parallel. The parallel time-stepping algorithm is illustrated for initial-boundary-value problems involving a linear advection-diffusion equation and the (nonlinear) reactive Euler equations. Numerical results are presented for both equations to demonstrate the accuracy and correctness of the parallel approach. Exact solutions of the advection-diffusion equation are constructed, and these are used to check the corresponding numerical solutions for a variety of tests involving different overlapping grids, different numbers of refinement levels and refinement ratios, and different numbersmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 941385

- Report Number(s):
- UCRL-JRNL-236681

Journal ID: ISSN 0021-9991; JCTPAH; TRN: US200824%%575

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics, vol. 227, no. 16, August 1, 2008, pp. 7469-7502

- Additional Journal Information:
- Journal Volume: 227; Journal Issue: 16; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS; 42 ENGINEERING; ACCURACY; ALGORITHMS; COMPUTERS; DIFFRACTION; EXACT SOLUTIONS; EXPLOSIONS; HOT SPOTS; IMPLEMENTATION; INTERPOLATION; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; PERFORMANCE; RESOLUTION

### Citation Formats

```
Henshaw, W, and Schwendeman, D.
```*Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement*. United States: N. p., 2007.
Web.

```
Henshaw, W, & Schwendeman, D.
```*Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement*. United States.

```
Henshaw, W, and Schwendeman, D. Thu .
"Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement". United States. https://www.osti.gov/servlets/purl/941385.
```

```
@article{osti_941385,
```

title = {Parallel Computation of Three-Dimensional Flows using Overlapping Grids with Adaptive Mesh Refinement},

author = {Henshaw, W and Schwendeman, D},

abstractNote = {This paper describes an approach for the numerical solution of time-dependent partial differential equations in complex three-dimensional domains. The domains are represented by overlapping structured grids, and block-structured adaptive mesh refinement (AMR) is employed to locally increase the grid resolution. In addition, the numerical method is implemented on parallel distributed-memory computers using a domain-decomposition approach. The implementation is flexible so that each base grid within the overlapping grid structure and its associated refinement grids can be independently partitioned over a chosen set of processors. A modified bin-packing algorithm is used to specify the partition for each grid so that the computational work is evenly distributed amongst the processors. All components of the AMR algorithm such as error estimation, regridding, and interpolation are performed in parallel. The parallel time-stepping algorithm is illustrated for initial-boundary-value problems involving a linear advection-diffusion equation and the (nonlinear) reactive Euler equations. Numerical results are presented for both equations to demonstrate the accuracy and correctness of the parallel approach. Exact solutions of the advection-diffusion equation are constructed, and these are used to check the corresponding numerical solutions for a variety of tests involving different overlapping grids, different numbers of refinement levels and refinement ratios, and different numbers of processors. The problem of planar shock diffraction by a sphere is considered as an illustration of the numerical approach for the Euler equations, and a problem involving the initiation of a detonation from a hot spot in a T-shaped pipe is considered to demonstrate the numerical approach for the reactive case. For both problems, the solutions are shown to be well resolved on the finest grid. The parallel performance of the approach is examined in detail for the shock diffraction problem.},

doi = {},

journal = {Journal of Computational Physics, vol. 227, no. 16, August 1, 2008, pp. 7469-7502},

issn = {0021-9991},

number = 16,

volume = 227,

place = {United States},

year = {2007},

month = {11}

}