Convergence Analysis for Spectral Approximation to a Scalar Transport Equation with a Random Wave Speed

Zhou, Tao; Tang, Tao

doi:10.4208/jcm.1206-m4012

Cited by 6 publications

(3 citation statements)

References 28 publications

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“…It is natural that some assumptions for the given data will be made. More precisely, we make the following assumptions (see [20,26]).…”

Section: Regularity Of the Discrete Solution In The Random Spacementioning

confidence: 99%

“…Such methods outperform the classical Monte-Carlo method in that, given sufficient regularity of the solution in the random space, they can achieve the spectral convergence, thus are much more efficient for problems with random uncertainties. Unfortunately, for hyperbolic problems, one often is not blessed with such regularities, which leads to significant reduction of order of convergence [17,26], thus slows down the computation or even gives non-convergent results due to Gibbs' phenomenon. The problems under study in this paper are problems with discontinuous solutions in the random space, due to jumps of solutions formed at the interfaces or barriers which will propagate into the random space.…”

Section: Introductionmentioning

confidence: 99%

“…In such problems the uncertainty may also come from the initial data. This is a well-studied problem [6,26] and our method can obviously be used in this case.…”

Section: Introductionmentioning

confidence: 99%

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The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

Jin

2017

J Sci Comput

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We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singular nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite difference or finite volume approximations in space and time for example, is smoother, we first discretize the equation by a smooth finite difference or finite volume scheme, and then use the gPC-SG approximation to the discrete system. The jump condition at the interface is treated using the immersed upwind methods introduced in [8,12]. This yields a method that converges with the spectral accuracy for finite mesh size and time step. We use a linear hyperbolic equation with discontinuous and random coefficient, and the Liouville equation with discontinuous and random potential, to illustrate our idea, with both one and second order spatial discretizations. Spectral convergence is established for the first equation, and numerical examples for both equations show the desired accuracy of the method.

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“…It is natural that some assumptions for the given data will be made. More precisely, we make the following assumptions (see [20,26]).…”

Section: Regularity Of the Discrete Solution In The Random Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

Jin

2017

J Sci Comput

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A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Rohde

Giesselmann

2019

IMA Journal of Numerical Analysis

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In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the Stochastic Galerkin method and for the spatial-temporal discretization of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016), this leads to computable error bounds for the space-stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.

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Hypocoercivity and Uniform Regularity for the Vlasov--Poisson--Fokker--Planck System with Uncertainty and Multiple Scales

Jin¹,

Zhu²

2018

SIAM J. Math. Anal.

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We study the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales.Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the initial data, the regularity of the solution in the random space, as well as exponential decay of the solution to the global Maxwellian, are established under Sobolev norms, which are uniform in terms of the scaling parameters. These are the first hypocoercivity results for a nonlinear kinetic system with random input, which are important for the understanding of the sensitivity of the system under random perturbations, and for the establishment of spectral convergence of popular numerical methods for uncertainty quantification based on (spectrally accurate) polynomial chaos expansions.

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Convergence Analysis for Spectral Approximation to a Scalar Transport Equation with a Random Wave Speed

Cited by 6 publications

References 28 publications

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

The Discrete Stochastic Galerkin Method for Hyperbolic Equations with Non-smooth and Random Coefficients

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Hypocoercivity and Uniform Regularity for the Vlasov--Poisson--Fokker--Planck System with Uncertainty and Multiple Scales

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