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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Carpenter MH, Gottlieb D, Abarbanel S (1994) Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J Comput Phys 111(2):220–236. doi:http://dx.doi.org/10.1006/jcph.1994.1057
Carpenter MH, Nordström J, Gottlieb D (1999) A stable and conservative interface treatment of arbitrary spatial accuracy. J Comput Phys 148(2):341–365. doi:http://dx.doi.org/10.1006/jcph.1998.6114
Chen QY, Gottlieb D, Hesthaven JS (2005) Uncertainty analysis for the steady-state flows in a dual throat nozzle. J Comput Phys 204(1):378–398. doi:http://dx.doi.org/10.1016/j.jcp.2004.10.019
Gottlieb D, Hesthaven JS (2001) Spectral methods for hyperbolic problems. J Comput Appl Math 128(1–2):83–131. doi:http://dx.doi.org/10.1016/S0377-0427(00)00510-0
Gottlieb D, Xiu D (2008) Galerkin method for wave equations with uncertain coefficients. Commun Comput Phys 3(2):505–518
Hou TY, Luo W, Rozovskii B, Zhou HM (2006) Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J Comput Phys 216(2):687–706. doi:http://dx.doi.org/10.1016/j.jcp.2006.01.008
LeVeque RJ (2006) Numerical methods for conservation laws, 2nd edn. Birkhäuser Basel
Mattsson K, Svärd M, Nordström J (2004) Stable and accurate artificial dissipation. J Sci Comput 21(1):57–79
Nordström J (2006) Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J Sci Comput 29(3):375–404. doi:http://dx.doi.org/10.1007/s10915-005-9013-4
Nordström J, Carpenter MH (1999) Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. J Comput Phys 148(2):621–645. doi:http://dx.doi.org/10.1006/jcph.1998.6133
Nordström J, Carpenter MH (2001) High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates. J Comput Phys 173(1):149–174. doi:http://dx.doi.org/10.1006/jcph.2001.6864
Pettersson P, Iaccarino G, Nordström J (2009) Numerical analysis of the Burgers’ equation in the presence of uncertainty. J Comput Phys 228:8394–8412. doi:10.1016/j.jcp.2009.08.012, http://dl.acm.org/citation.cfm?id=1621150.1621394
Richtmyer RD, Morton KW (1967) Difference methods for initial-value problems, 1st edn. Interscience, New York
Xiu D, Karniadakis GE (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167. doi:10.1016/S0021-9991(03)00092-5, http://dx.doi.org/10.1016/S0021-9991(03)00092-5
Xiu D, Karniadakis GE (2004) Supersensitivity due to uncertain boundary conditions. Int J Numer Meth Eng 61:2114–2138
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Pettersson, M.P., Iaccarino, G., Nordström, J. (2015). Nonlinear Transport Under Uncertainty. In: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-10714-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-10714-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10713-4
Online ISBN: 978-3-319-10714-1
eBook Packages: EngineeringEngineering (R0)