Uncertainty Quantification for Hyperbolic Systems of Conservation Laws

We review uncertainty quantification (UQ) for hyperbolic systems of conservation (balance) laws. The input uncertainty could be in the initial data, fluxes, coefficients, source terms or boundary conditions. We focus on forward UQ or uncertainty propagation and review deterministic methods such as stochastic Galerkin and stochastic collocation finite volume methods for approximating random (field) entropy solutions. Statistical sampling methods of the Monte Carlo and multilevel Monte Carlo (MLMC) type are also described. We present alternative UQ frameworks such as measure-valued solutions and statistical solutions.

[1]  Florian Müller,et al.  Multilevel Monte Carlo for two phase flow and Buckley-Leverett transport in random heterogeneous porous media , 2013, J. Comput. Phys..

[2]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[3]  Eitan Tadmor,et al.  On the computation of measure-valued solutions , 2016, Acta Numerica.

[4]  Siddhartha Mishra,et al.  Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations , 2011 .

[5]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[6]  Chi-Wang Shu,et al.  High order time discretization methods with the strong stability property , 2001 .

[7]  Rémi Abgrall,et al.  Sensitivity analysis and characterization of the uncertain input data for the EXPERT vehicle , 2013 .

[8]  Siddhartha Mishra,et al.  Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data , 2012, Math. Comput..

[9]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[10]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[11]  A. Harten.  High Resolution Schemes for Hyperbolic Conservation Laws , 2017 .

[12]  G E Karniadakis,et al.  The stochastic piston problem. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

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

[14]  P. G. Ciarlet,et al.  General lagrange and hermite interpolation in Rn with applications to finite element methods , 1972 .

[15]  A. Harten.  Multiresolution algorithms for the numerical solution of hyperbolic conservation laws , 2010 .

[16]  Xiu Yang,et al.  Adaptive ANOVA decomposition of stochastic incompressible and compressible flows , 2012, J. Comput. Phys..

[17]  Siddhartha Mishra,et al.  Statistical solutions of hyperbolic systems of conservation laws: numerical approximation , 2019, Mathematical Models and Methods in Applied Sciences.

[18]  M. Chial,et al.  in simple , 2003 .

[19]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[20]  Timothy J. Barth,et al.  On the propagation of statistical model parameter uncertainty in CFD calculations , 2012 .

[21]  Svetlana Tokareva,et al.  High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data , 2011 .

[22]  Pietro Marco Congedo,et al.  Bayesian-based method with metamodels for rebuilding freestream conditions in atmospheric entry flows , 2013 .

[23]  Manuel J. Castro,et al.  Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis , 2015 .

[24]  S. Savage,et al.  The motion of a finite mass of granular material down a rough incline , 1989, Journal of Fluid Mechanics.

[25]  Alexandre Ern,et al.  Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems , 2010, J. Comput. Phys..

[26]  Alexandre Ern,et al.  Roe solver with entropy corrector for uncertain hyperbolic systems , 2010, J. Comput. Appl. Math..

[27]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[28]  Rémi Abgrall,et al.  Robust model reduction by $$L^{1}$$L1-norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems , 2016, Adv. Model. Simul. Eng. Sci..

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

[30]  Jonas Sukys,et al.  Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws , 2013, Uncertainty Quantification in Computational Fluid Dynamics.

[31]  Rémi Abgrall,et al.  A one-time truncate and encode multiresolution stochastic framework , 2014, J. Comput. Phys..

[32]  Jan S. Hesthaven,et al.  Uncertainty analysis for the steady-state flows in a dual throat nozzle , 2005 .

[33]  Alexander Kurganov,et al.  A well-balanced operator splitting based stochastic Galerkin method for the one-dimensional Saint-Venant system with uncertainty , 2015 .

[34]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

[35]  C. Schwab,et al.  Multi-level Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions , 2011 .

[36]  R. Tempone,et al.  A continuation multilevel Monte Carlo algorithm , 2014, BIT Numerical Mathematics.

[37]  Jonas Sukys,et al.  Multilevel Monte Carlo Finite Volume Methods for Shallow Water Equations with Uncertain Topography in Multi-dimensions , 2012, SIAM J. Sci. Comput..

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

[39]  Eitan Tadmor,et al.  Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography , 2011, J. Comput. Phys..

[40]  I. Müller,et al.  Rational Extended Thermodynamics , 1993 .

[41]  Timothy J. Barth,et al.  Non-intrusive Uncertainty Propagation with Error Bounds for Conservation Laws Containing Discontinuities , 2013, Uncertainty Quantification in Computational Fluid Dynamics.

[42]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[43]  Rémi Abgrall,et al.  Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes , 2011, J. Comput. Phys..

[44]  Alexander Kurganov,et al.  An operator splitting based stochastic Galerkin method for the one-dimensional compressible Euler equations with uncertainty , 2015 .

[45]  R. Abgrall,et al.  Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations , 2015, J. Comput. Phys..

[46]  Michael B. Giles,et al.  Multilevel Monte Carlo methods , 2013, Acta Numerica.

[47]  Rémi Abgrall,et al.  On the use of Mühlbach expansions in the recovery step of ENO methods , 1997 .

[48]  Jeroen A. S. Witteveen,et al.  Essentially Non-oscillatory Stencil Selection and Subcell Resolution in Uncertainty Quantification , 2013, Uncertainty Quantification in Computational Fluid Dynamics.

[49]  R. J. Diperna,et al.  Measure-valued solutions to conservation laws , 1985 .

[50]  F. Müller.  Stochastic methods for uncertainty quantification in subsurface flow and transport problems , 2014 .

[51]  Rémi Abgrall,et al.  Residual distribution schemes: Current status and future trends , 2006 .

[52]  Siddhartha Mishra,et al.  Multi-level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems , 2013 .

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

[54]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[55]  Jonas Sukys,et al.  Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium , 2014, J. Comput. Phys..

[56]  Kjetil Olsen Lye,et al.  Multilevel Monte-Carlo for measure valued solutions , 2016, 1611.07732.

[57]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[58]  Eitan Tadmor,et al.  Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws , 2014, Found. Comput. Math..

[59]  Siddhartha Mishra,et al.  Statistical Solutions of Hyperbolic Conservation Laws: Foundations , 2016, Archive for Rational Mechanics and Analysis.

[60]  Jonas Šukys.  Robust multi-level Monte Carlo finite volume methods for systems of hyperbolic conservation laws with random input data , 2014 .

[61]  Hester Bijl,et al.  Uncertainty Quantification in Computational Fluid Dynamics , 2013, Lecture Notes in Computational Science and Engineering.

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

[63]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[64]  Siddhartha Mishra,et al.  Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws , 2014, Numerische Mathematik.

[65]  Rémi Abgrall,et al.  Stochastic Discrete Equation Method (sDEM) for two-phase flows , 2015, J. Comput. Phys..

[66]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

[67]  Jonas Sukys,et al.  Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions , 2012, J. Comput. Phys..

[68]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[69]  Eitan Tadmor,et al.  Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws , 2012, SIAM J. Numer. Anal..

[70]  Siddhartha Mishra,et al.  Static load balancing for multi-level Monte Carlo finite volume solvers , 2011 .

[71]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[72]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[73]  P. Frederickson,et al.  Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction , 1990 .

[74]  J. Sukys.  Adaptive Load Balancing for Massively Parallel Multi-Level Monte Carlo Solvers , 2013, PPAM.

[75]  Svetlana Tokareva,et al.  Numerical Solution of Scalar Conservation Laws with Random Flux Functions , 2016, SIAM/ASA J. Uncertain. Quantification.

[76]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[77]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[78]  Rémi Abgrall,et al.  A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems , 2013, J. Comput. Phys..

[79]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[80]  Manuel Jesús Castro Díaz,et al.  On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System , 2011, J. Sci. Comput..

[81]  Dante De Santis,et al.  High-order linear and non-linear residual distribution schemes for turbulent compressible flows , 2015 .

[82]  G. Karniadakis,et al.  Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations , 2006 .

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

[84]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .

[85]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[86]  Rémi Abgrall,et al.  A simple semi-intrusive method for Uncertainty Quantification of shocked flows, comparison with a non-intrusive Polynomial Chaos method , 2010 .

[87]  J. Glimm,et al.  Chaos, transport and mesh convergence for fluid mixing , 2008 .