# A dynamical polynomial chaos approach for long-time evolution of SPDEs

## Abstract

We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen–Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier–Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22701587

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 343; Other Information: Copyright (c) 2017 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; CHAOS THEORY; COMPUTERIZED SIMULATION; MARKOV PROCESS; MATHEMATICAL EVOLUTION; MATHEMATICAL SOLUTIONS; MONTE CARLO METHOD; NAVIER-STOKES EQUATIONS; NOISE; POLYNOMIALS; TIME DEPENDENCE

### Citation Formats

```
Ozen, H. Cagan, E-mail: hco2104@columbia.edu, and Bal, Guillaume.
```*A dynamical polynomial chaos approach for long-time evolution of SPDEs*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.04.054.

```
Ozen, H. Cagan, E-mail: hco2104@columbia.edu, & Bal, Guillaume.
```*A dynamical polynomial chaos approach for long-time evolution of SPDEs*. United States. doi:10.1016/J.JCP.2017.04.054.

```
Ozen, H. Cagan, E-mail: hco2104@columbia.edu, and Bal, Guillaume. Tue .
"A dynamical polynomial chaos approach for long-time evolution of SPDEs". United States. doi:10.1016/J.JCP.2017.04.054.
```

```
@article{osti_22701587,
```

title = {A dynamical polynomial chaos approach for long-time evolution of SPDEs},

author = {Ozen, H. Cagan, E-mail: hco2104@columbia.edu and Bal, Guillaume},

abstractNote = {We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen–Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier–Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.},

doi = {10.1016/J.JCP.2017.04.054},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 343,

place = {United States},

year = {2017},

month = {8}

}