Numerical solutions of stochastic PDEs driven by arbitrary type of noise

Chen, Tianheng; Rozovskiĭ, B. L.; Shu, Chi-Wang

doi:10.1007/s40072-018-0120-2

Cited by 2 publications

(2 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years, numerous studies have been focused on advanced and efficient methods for SPDEs such as finite difference methods [16,21,22,33,42,45], finite element methods [1,17,19,28,44,47], spectral methods [25,31,34,35], and also some other types of numerical methods [7,41]. Concerning discontinuous finite element methods for SPDEs, Cao et al [4,5] developed a discontinuous Galerkin (DG) method to the time-independent elliptic SPDEs with additive noises.…”

Section: Introductionmentioning

confidence: 99%

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

Shu

Tang

2021

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

In this paper, we propose a local discontinuous Galerkin (LDG) method for fully nonlinear and possibly degenerate parabolic stochastic partial differential equations (SPDEs), which is a high-order numerical scheme. This method is an extension of the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and share with the DG method its advantage and flexibility. We prove the $L^2$-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ($\cO(h^{k+1})$) for smooth solutions of semilinear stochastic equations is shown if polynomials of degree $k$ are used. We also develop an explicit derivative-free order $1.5$ time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples displaying the performance of the method are shown.

show abstract

Section: Introductionmentioning

confidence: 99%

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

Shu

Tang

2021

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

show abstract

“…Stochastic differential equations are being used for modelling hydrological systems for a long time (Gupta, Bhattacharya, & Sposito, 1981) and still the researchers are actively involved to find the optimal solution strategies to deal with irregularities (Asmuth & Bierkens, 2005;D'Odorico, Laio, & Ridolfi, 2005;Peterson & Western, 2014). These SPDEs are either driven by space-time Brownian motion (Kalpinelli, Frangos, & Yannacopoulos, 2011;Kulasiri, 2013) or more generally, space-time Levy process (Arne, Oksendal, & Frank, 2004;Chen, Rozovskii, & A Shu, 2019) and solved analytically as well as numerically in several monographs and research articles. Brownian motion is defined as the irregular movement of particles suspended in a fluid.…”

Section: Introductionmentioning

confidence: 99%

Role of Wiener chaos expansion in modelling randomness for groundwater contamination flow

Tiwari¹,

Kulasiri²,

Samarasinghe

2019

El Sawah, S. (Ed.) MODSIM2019, 23rd International Congress on Modelling and Simulation.

View full text Add to dashboard Cite

Pertinent mathematical modelling plays pivot role in making groundwater protection and reclamation policies. Uncertain parameters and several basic phenomena in almost all branches of engineering and science can be modelled efficiently with the help of Stochastic Partial Differential Equations (SPDEs) and their behaviour can be interpreted more accurately. The intent of the present study is to use an efficient numerical approach based on Wiener chaos expansion to understand the stochastic nature of variables associated with groundwater flow. First and second order moments of concentration profile are calculated and plotted graphically. Obtained results are in good aggrement with those available in existing literature.

show abstract

Numerical solutions of stochastic PDEs driven by arbitrary type of noise

Cited by 2 publications

References 30 publications

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

Role of Wiener chaos expansion in modelling randomness for groundwater contamination flow

Contact Info

Product

Resources

About