Intrusive projection methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in physical as well as in stochastic spaces. The stochastic spectral method relies on multi-resolution schemes where the stochastic space is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization in physical space and time. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.

[1]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[2]  Y. Marzouk,et al.  Uncertainty quantification in chemical systems , 2009 .

[3]  O. L. Maître,et al.  Uncertainty propagation in CFD using polynomial chaos decomposition , 2006 .

[4]  George E. Karniadakis,et al.  The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications , 2008, J. Comput. Phys..

[5]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

[6]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[7]  Baskar Ganapathysubramanian,et al.  Sparse grid collocation schemes for stochastic natural convection problems , 2007, J. Comput. Phys..

[8]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[9]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[10]  Marcelo H. Kobayashi,et al.  Stochastic Solution for Uncertainty Propagation in Nonlinear Shallow-Water Equations , 2008 .

[11]  A. Chorin.  Gaussian fields and random flow , 1974, Journal of Fluid Mechanics.

[12]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[13]  N. Wiener.  The Homogeneous Chaos , 1938 .

[14]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[15]  Rémi Abgrall.  A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems , 2008 .

[16]  Hermann G. Matthies,et al.  Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations , 2003 .

[17]  H. Najm,et al.  Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection , 2003 .

[18]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[19]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[20]  R. Ghanem,et al.  A stochastic projection method for fluid flow. I: basic formulation , 2001 .

[21]  Habib N. Najm,et al.  Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions , 2005, SIAM J. Sci. Comput..

[22]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[23]  Bruno Després,et al.  Uncertainty quantification for systems of conservation laws , 2009, J. Comput. Phys..

[24]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[25]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[26]  George Em Karniadakis,et al.  Stochastic modeling of random roughness in shock scattering problems: Theory and simulations , 2008 .

[27]  O. L. Maître,et al.  Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation , 2003 .

[28]  M. Ndjinga.  Computing the matrix sign and absolute value functions , 2008 .

[29]  Dongbin Xiu,et al.  Galerkin method for wave equations with uncertain coefficients , 2008 .

[30]  Thomas A. Zang,et al.  Stochastic approaches to uncertainty quantification in CFD simulations , 2005, Numerical Algorithms.

[31]  Habib N. Najm,et al.  Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems , 2007, SIAM J. Sci. Comput..

[32]  Guang Lin,et al.  Predicting shock dynamics in the presence of uncertainties , 2006, J. Comput. Phys..

[33]  O. L. Maître,et al.  A Newton method for the resolution of steady stochastic Navier–Stokes equations , 2009 .

[34]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[35]  Pierre Degond,et al.  Polynomial upwind schemes for hyperbolic systems , 1999 .

[36]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[37]  L. Mathelin,et al.  A Stochastic Collocation Algorithm for Uncertainty Analysis , 2003 .