# Parallel adaptive wavelet collocation method for PDEs

## Abstract

A parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogeneous turbulence with linear forcing at effective non-adaptive resolutions up to 2048{sup 3} using as many as 2048 CPU cores.

- Authors:

- FortiVenti Inc., Suite 404, 999 Canada Place, Vancouver, BC, V6C 3E2 (Canada)
- Department of Mechanical Engineering, University of Colorado Boulder, UCB 427, Boulder, CO 80309 (United States)

- Publication Date:

- OSTI Identifier:
- 22465660

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 298; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; DYNAMIC LOADS; PARTIAL DIFFERENTIAL EQUATIONS; PARTITION; QUANTUM MECHANICS; RESOLUTION; SIMULATION; SYNCHRONIZATION; TURBULENCE

### Citation Formats

```
Nejadmalayeri, Alireza, E-mail: Alireza.Nejadmalayeri@gmail.com, Vezolainen, Alexei, E-mail: Alexei.Vezolainen@Colorado.edu, Brown-Dymkoski, Eric, E-mail: Eric.Browndymkoski@Colorado.edu, and Vasilyev, Oleg V., E-mail: Oleg.Vasilyev@Colorado.edu.
```*Parallel adaptive wavelet collocation method for PDEs*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2015.05.028.

```
Nejadmalayeri, Alireza, E-mail: Alireza.Nejadmalayeri@gmail.com, Vezolainen, Alexei, E-mail: Alexei.Vezolainen@Colorado.edu, Brown-Dymkoski, Eric, E-mail: Eric.Browndymkoski@Colorado.edu, & Vasilyev, Oleg V., E-mail: Oleg.Vasilyev@Colorado.edu.
```*Parallel adaptive wavelet collocation method for PDEs*. United States. doi:10.1016/J.JCP.2015.05.028.

```
Nejadmalayeri, Alireza, E-mail: Alireza.Nejadmalayeri@gmail.com, Vezolainen, Alexei, E-mail: Alexei.Vezolainen@Colorado.edu, Brown-Dymkoski, Eric, E-mail: Eric.Browndymkoski@Colorado.edu, and Vasilyev, Oleg V., E-mail: Oleg.Vasilyev@Colorado.edu. Thu .
"Parallel adaptive wavelet collocation method for PDEs". United States. doi:10.1016/J.JCP.2015.05.028.
```

```
@article{osti_22465660,
```

title = {Parallel adaptive wavelet collocation method for PDEs},

author = {Nejadmalayeri, Alireza, E-mail: Alireza.Nejadmalayeri@gmail.com and Vezolainen, Alexei, E-mail: Alexei.Vezolainen@Colorado.edu and Brown-Dymkoski, Eric, E-mail: Eric.Browndymkoski@Colorado.edu and Vasilyev, Oleg V., E-mail: Oleg.Vasilyev@Colorado.edu},

abstractNote = {A parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogeneous turbulence with linear forcing at effective non-adaptive resolutions up to 2048{sup 3} using as many as 2048 CPU cores.},

doi = {10.1016/J.JCP.2015.05.028},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 298,

place = {United States},

year = {2015},

month = {10}

}