Sensitivity analysis and stochastic simulations of non‐equilibrium plasma flow

We study the parametric uncertainties involved in plasma flows and apply stochastic sensitivity analysis to rank the importance of all inputs to guide large-scale stochastic simulations. Specifically, we employ different gradient-based sensitivity methods, namely Morris, multi-element probabilistic collocation method on sparse grids, Quasi-Monte Carlo and Monte Carlo methods. These approaches go beyond the standard ‘One-At-a-Time’ sensitivity analysis and provide a measure of the non-linear interaction effects for the uncertain inputs. The objective is to perform systematic stochastic simulations of plasma flows treating only as stochastic processes the inputs with the highest sensitivity index, hence reducing substantially the computational cost. Two plasma flow examples are presented to demonstrate the capability and efficiency of the stochastic sensitivity analysis. The first one is a two-fluid model in a shock tube whereas the second one is a one-fluid/two-temperature model in flow past a cylinder. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  George Em Karniadakis,et al.  Random roughness enhances lift in supersonic flow. , 2007, Physical review letters.

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

[3]  Jefferson W. Tester,et al.  Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water , 1998 .

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

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

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

[7]  Didier Lucor,et al.  Covariance kernel representations of multidimensional second-order stochastic processes , 2006, J. Comput. Phys..

[8]  Max D. Morris,et al.  Factorial sampling plans for preliminary computational experiments , 1991 .

[9]  S. Isukapalli,et al.  Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems , 1998, Risk analysis : an official publication of the Society for Risk Analysis.

[10]  Guang Lin,et al.  Stochastic Computational Fluid Mechanics , 2007, Computing in Science & Engineering.

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

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

[13]  Arne Dür,et al.  On the Optimality of the Discrete Karhunen--Loève Expansion , 1998 .

[14]  R. Ghanem,et al.  Stochastic Finite-Element Analysis of Seismic Soil-Structure Interaction , 2002 .

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

[16]  R. Cools.  Monomial cubature rules since “Stroud”: a compilation—part 2 , 1999 .

[17]  M. Kramer,et al.  Sensitivity Analysis in Chemical Kinetics , 1983 .

[18]  R. Ghanem,et al.  Quantifying uncertainty in chemical systems modeling , 2004 .

[19]  Menner A. Tatang,et al.  An efficient method for parametric uncertainty analysis of numerical geophysical models , 1997 .

[20]  R. Walters.  Stochastic Fluid Mechanics via Polynomial Chaos , 2003 .

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

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

[23]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .

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

[25]  Uri Shumlak,et al.  Approximate Riemann solver for the two-fluid plasma model , 2003 .

[26]  Andrei P. Sokolov,et al.  A Methodology for Quantifying Uncertainty in Climate Projections , 2000 .

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

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

[29]  S. C. Cotter.  A screening design for factorial experiments with interactions , 1979 .

[30]  A. Quarteroni,et al.  Approximation results for orthogonal polynomials in Sobolev spaces , 1982 .

[31]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

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

[33]  Pradya Prempraneerach,et al.  Uncertainty analysis in a shipboard integrated power system using multi-element polynomial chaos , 2007 .

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

[35]  Wim Schoutens,et al.  Enhancing the Morris Method , 2005 .

[36]  A discontinuous Galerkin method for two-temperature plasmas , 2006 .

[37]  Christine Bernardi,et al.  Properties of some weighted Sobolev spaces and application to spectral approximations , 1989 .

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

[39]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[40]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[41]  Bernardus Wilhelmus Maria Bettonvil,et al.  Detection of important factors by sequential bifurcation , 1990 .