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Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsNonlinear Transport Under Uncertainty

Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Nonlinear Transport Under... [Burgers’ equation is an interesting and non-linear model problem from which, many results can be extended to other hyperbolic systems, e.g., the Euler equations. In this chapter, a detailed uncertainty quantification analysis is performed for the Burgers’ equation; we employ a spectral representation of the solution in the form of polynomial chaos expansion. The equation is stochastic as a result of the uncertainty in the initial and boundary values. Galerkin projection results in a coupled, deterministic system of hyperbolic equations from which statistics of the solution can be determined. A well-posed stochastic Galerkin formulation is presented and a strongly stable numerical scheme is devised. The effect of missing data is investigated, in terms of both stability of the numerical scheme and accuracy of the numerical solution.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsNonlinear Transport Under Uncertainty

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-10713-4
Pages
81 –109
DOI
10.1007/978-3-319-10714-1_6
Publisher site
See Chapter on Publisher Site

Abstract

[Burgers’ equation is an interesting and non-linear model problem from which, many results can be extended to other hyperbolic systems, e.g., the Euler equations. In this chapter, a detailed uncertainty quantification analysis is performed for the Burgers’ equation; we employ a spectral representation of the solution in the form of polynomial chaos expansion. The equation is stochastic as a result of the uncertainty in the initial and boundary values. Galerkin projection results in a coupled, deterministic system of hyperbolic equations from which statistics of the solution can be determined. A well-posed stochastic Galerkin formulation is presented and a strongly stable numerical scheme is devised. The effect of missing data is investigated, in terms of both stability of the numerical scheme and accuracy of the numerical solution.]

Published: Sep 17, 2014

Keywords: Riemann Problem; Shock Speed; Polynomial Chaos; Polynomial Chaos Expansion; Dissipation Operator

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