# Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance

## Abstract

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Here, our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of California, Santa Barbara, CA (United States)

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

- OSTI Identifier:
- 1562583

- Alternate Identifier(s):
- OSTI ID: 1574443; OSTI ID: 1633911

- Report Number(s):
- SAND-2019-12136J

Journal ID: ISSN 2590-0374; S2590037419300688; 100068; PII: S2590037419300688

- Grant/Contract Number:
- SC0009254; AC04-94AL85000

- Resource Type:
- Published Article

- Journal Name:
- Results in Applied Mathematics

- Additional Journal Information:
- Journal Name: Results in Applied Mathematics Journal Volume: 4 Journal Issue: C; Journal ID: ISSN 2590-0374

- Publisher:
- Elsevier

- Country of Publication:
- Netherlands

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Stochastic partial differential equations; Fluctuation-dissipation balance; Discontinuous Galerkin methods

### Citation Formats

```
Pazner, W., Trask, N., and Atzberger, P. J. Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance. Netherlands: N. p., 2019.
Web. doi:10.1016/j.rinam.2019.100068.
```

```
Pazner, W., Trask, N., & Atzberger, P. J. Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance. Netherlands. doi:10.1016/j.rinam.2019.100068.
```

```
Pazner, W., Trask, N., and Atzberger, P. J. Sun .
"Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance". Netherlands. doi:10.1016/j.rinam.2019.100068.
```

```
@article{osti_1562583,
```

title = {Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance},

author = {Pazner, W. and Trask, N. and Atzberger, P. J.},

abstractNote = {We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Here, our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.},

doi = {10.1016/j.rinam.2019.100068},

journal = {Results in Applied Mathematics},

number = C,

volume = 4,

place = {Netherlands},

year = {2019},

month = {12}

}

DOI: 10.1016/j.rinam.2019.100068