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

TL;DR

The paper introduces a Dynamical generalized Polynomial Chaos (DgPC) method that enables efficient long-time simulation of SPDEs with white noise forcing by using restart procedures and orthogonal polynomial expansions.

Contribution

It presents a novel DgPC approach that maintains low stochastic dimensionality over time, allowing for long-term SPDE simulations including invariant measures.

Findings

DgPC effectively simulates long-time SPDE solutions.

The method compares favorably with Monte Carlo simulations.

It can incorporate time-independent random coefficients.

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 [1] 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.