A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations the first author would like to dedicate this paper to his Father, Ahmed Baccouch, who unfortunately passed away during the completion of this work. Without his father's support and encouragement, he definitely would not become a professor of mathematics

Mahboub Baccouch, Bryan Johnson

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5 Citations (Scopus)

Abstract

In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of Itô stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary differential equations (ODEs), we first construct an approximate deterministic ODE with a random coefficient on each element using the well-known Wong-Zakai approximation theorem. Since the resulting ODE converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain a deterministic ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard DG method for deterministic ODEs. We prove that the proposed stochastic DG (SDG) method is equivalent to an implicit stochastic Runge-Kutta method. Then, we study the numerical stability of the SDG scheme applied to linear SDEs with an additive noise term. The method is shown to be numerically stable in the mean sense and also A-stable. As a result, it is suitable for solving stiff SDEs. Moreover, the method is proved to be convergent in the mean-square sense. Numerical evidence demonstrates that our proposed DG scheme for SDEs with additive noise has a strong convergence order of 2p+1, when degree piecewise polynomials are used. When applied to SDEs with multiplicative noise, the SDG method is strongly convergent with order. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems.

Original languageEnglish (US)
Pages (from-to)138-165
Number of pages28
JournalJournal of Computational and Applied Mathematics
Volume308
DOIs
StatePublished - Dec 15 2016

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Stochastic Ordinary Differential Equations
Discontinuous Galerkin Method
Galerkin methods
Ordinary differential equations
Completion
Higher Order
Ordinary differential equation
Additive Noise
Stiff Ordinary Differential Equations
Additive noise
Converge
Convergence Order
Random Coefficients
Linear Ordinary Differential Equations
Discontinuous Galerkin
Approximation Theorem
Multiplicative Noise
Piecewise Polynomials
Wiener Process
Stochastic Methods

Keywords

  • A-stability
  • Discontinuous Galerkin method
  • It stochastic differential equation
  • Mean-square convergence
  • Order of convergence
  • WongZakai approximation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{6e9b756e2abe4d069d61eb658737be28,
title = "A high-order discontinuous Galerkin method for It{\^o} stochastic ordinary differential equations the first author would like to dedicate this paper to his Father, Ahmed Baccouch, who unfortunately passed away during the completion of this work. Without his father's support and encouragement, he definitely would not become a professor of mathematics",
abstract = "In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of It{\^o} stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary differential equations (ODEs), we first construct an approximate deterministic ODE with a random coefficient on each element using the well-known Wong-Zakai approximation theorem. Since the resulting ODE converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain a deterministic ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard DG method for deterministic ODEs. We prove that the proposed stochastic DG (SDG) method is equivalent to an implicit stochastic Runge-Kutta method. Then, we study the numerical stability of the SDG scheme applied to linear SDEs with an additive noise term. The method is shown to be numerically stable in the mean sense and also A-stable. As a result, it is suitable for solving stiff SDEs. Moreover, the method is proved to be convergent in the mean-square sense. Numerical evidence demonstrates that our proposed DG scheme for SDEs with additive noise has a strong convergence order of 2p+1, when degree piecewise polynomials are used. When applied to SDEs with multiplicative noise, the SDG method is strongly convergent with order. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems.",
keywords = "A-stability, Discontinuous Galerkin method, It stochastic differential equation, Mean-square convergence, Order of convergence, WongZakai approximation",
author = "Mahboub Baccouch and Bryan Johnson",
year = "2016",
month = "12",
day = "15",
doi = "10.1016/j.cam.2016.05.034",
language = "English (US)",
volume = "308",
pages = "138--165",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",

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TY - JOUR

T1 - A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations the first author would like to dedicate this paper to his Father, Ahmed Baccouch, who unfortunately passed away during the completion of this work. Without his father's support and encouragement, he definitely would not become a professor of mathematics

AU - Baccouch, Mahboub

AU - Johnson, Bryan

PY - 2016/12/15

Y1 - 2016/12/15

N2 - In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of Itô stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary differential equations (ODEs), we first construct an approximate deterministic ODE with a random coefficient on each element using the well-known Wong-Zakai approximation theorem. Since the resulting ODE converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain a deterministic ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard DG method for deterministic ODEs. We prove that the proposed stochastic DG (SDG) method is equivalent to an implicit stochastic Runge-Kutta method. Then, we study the numerical stability of the SDG scheme applied to linear SDEs with an additive noise term. The method is shown to be numerically stable in the mean sense and also A-stable. As a result, it is suitable for solving stiff SDEs. Moreover, the method is proved to be convergent in the mean-square sense. Numerical evidence demonstrates that our proposed DG scheme for SDEs with additive noise has a strong convergence order of 2p+1, when degree piecewise polynomials are used. When applied to SDEs with multiplicative noise, the SDG method is strongly convergent with order. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems.

AB - In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of Itô stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary differential equations (ODEs), we first construct an approximate deterministic ODE with a random coefficient on each element using the well-known Wong-Zakai approximation theorem. Since the resulting ODE converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain a deterministic ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard DG method for deterministic ODEs. We prove that the proposed stochastic DG (SDG) method is equivalent to an implicit stochastic Runge-Kutta method. Then, we study the numerical stability of the SDG scheme applied to linear SDEs with an additive noise term. The method is shown to be numerically stable in the mean sense and also A-stable. As a result, it is suitable for solving stiff SDEs. Moreover, the method is proved to be convergent in the mean-square sense. Numerical evidence demonstrates that our proposed DG scheme for SDEs with additive noise has a strong convergence order of 2p+1, when degree piecewise polynomials are used. When applied to SDEs with multiplicative noise, the SDG method is strongly convergent with order. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems.

KW - A-stability

KW - Discontinuous Galerkin method

KW - It stochastic differential equation

KW - Mean-square convergence

KW - Order of convergence

KW - WongZakai approximation

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