TY - JOUR

T1 - A stochastic Galerkin method for the Euler equations with Roe variable transformation

AU - Pettersson, Per

AU - Iaccarino, Gianluca

AU - Nordström, Jan

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work is supported by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2014/1

Y1 - 2014/1

N2 - The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc.

AB - The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc.

UR - http://hdl.handle.net/10754/597412

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999113006797

UR - http://www.scopus.com/inward/record.url?scp=84886783939&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2013.10.011

DO - 10.1016/j.jcp.2013.10.011

M3 - Article

VL - 257

SP - 481

EP - 500

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -