Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Harmon, Jake J.; Tokareva, Svetlana; Zlotnik, Anatoly; Swart, Pieter J.

doi:10.1137/23m1624750

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Journal Article · Wed Apr 02 00:00:00 EDT 2025 · SIAM/ASA Journal on Uncertainty Quantification

DOI:https://doi.org/10.1137/23m1624750· OSTI ID:2556812

^[1]; Tokareva, Svetlana ^[1]; Zlotnik, Anatoly ^[1]; Swart, Pieter J. ^[1]

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Here, we propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for nonlinear hyperbolic PDEs based on Burgers’ and Euler’s equations.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2556812

Report Number(s):: LA-UR--24-30309

Journal Information:: SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 2 Vol. 13; ISSN 2166-2525

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (48)

Efficient Implementation of Weighted ENO Schemes Jiang, Guang-Shan; Shu, Chi-Wang Journal of Computational Physics, Vol. 126, Issue 1 https://doi.org/10.1006/jcph.1996.0130	journal	June 1996
Non-intrusive Uncertainty Propagation with Error Bounds for Conservation Laws Containing Discontinuities Barth, Timothy Uncertainty Quantification in Computational Fluid Dynamics https://doi.org/10.1007/978-3-319-00885-1_1	book	January 2013
High Order SFV and Mixed SDG/FV Methods for the Uncertainty Quantification in Multidimensional Conservation Laws Tokareva, Svetlana; Schwab, Christoph; Mishra, Siddhartha Lecture Notes in Computational Science and Engineering https://doi.org/10.1007/978-3-319-05455-1_7	book	January 2014
Theh,p andh-p versions of the finite element method in 1 dimension: Part I. the error analysis of thep-version Gui, W.; Babuška, I. Numerische Mathematik, Vol. 49, Issue 6 https://doi.org/10.1007/BF01389733	journal	November 1986
Theh, p andh-p versions of the finite element method in 1 dimension Gui, W.; Babuška, I. Numerische Mathematik, Vol. 49, Issue 6 https://doi.org/10.1007/BF01389734	journal	November 1986
Theh, p andh-p versions of the finite element method in 1 dimension Gui, W.; Babuška, I. Numerische Mathematik, Vol. 49, Issue 6 https://doi.org/10.1007/BF01389735	journal	November 1986
Error estimates for the combinedh andp versions of the finite element method Babuška, Ivo; Dorr, Milo R. Numerische Mathematik, Vol. 37, Issue 2 https://doi.org/10.1007/BF01398256	journal	June 1981
Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws Shu, Chi-Wang Lecture Notes in Mathematics https://doi.org/10.1007/BFb0096355	book	January 1998
Riemann Solvers and Numerical Methods for Fluid Dynamics Toro, Eleuterio F. https://doi.org/10.1007/b79761	book	January 2009
Adjoint Methods for Guiding Adaptive Mesh Refinement in Tsunami Modeling Davis, B. N.; LeVeque, R. J. Pure and Applied Geophysics, Vol. 173, Issue 12 https://doi.org/10.1007/s00024-016-1412-y	journal	October 2016
Error estimation and uncertainty quantification for first time to a threshold value Chaudhry, Jehanzeb H.; Estep, Donald; Stevens, Zachary BIT Numerical Mathematics, Vol. 61, Issue 1 https://doi.org/10.1007/s10543-020-00825-0	journal	August 2020
Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case Gerhard, Nils; Müller, Siegfried Computational and Applied Mathematics, Vol. 35, Issue 2 https://doi.org/10.1007/s40314-014-0134-y	journal	April 2014
Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method van Leer, Bram Journal of Computational Physics, Vol. 32, Issue 1 https://doi.org/10.1016/0021-9991(79)90145-1	journal	July 1979
Adaptive mesh refinement for hyperbolic partial differential equations Berger, Marsha J.; Oliger, Joseph Journal of Computational Physics, Vol. 53, Issue 3 https://doi.org/10.1016/0021-9991(84)90073-1	journal	March 1984
The numerical simulation of two-dimensional fluid flow with strong shocks Woodward, Paul; Colella, Phillip Journal of Computational Physics, Vol. 54, Issue 1 https://doi.org/10.1016/0021-9991(84)90142-6	journal	April 1984
Local adaptive mesh refinement for shock hydrodynamics Berger, M. J.; Colella, P. Journal of Computational Physics, Vol. 82, Issue 1 https://doi.org/10.1016/0021-9991(89)90035-1	journal	May 1989
Modeling uncertainty in flow simulations via generalized polynomial chaos Xiu, Dongbin; Karniadakis, George Em Journal of Computational Physics, Vol. 187, Issue 1 https://doi.org/10.1016/S0021-9991(03)00092-5	journal	May 2003
Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks Tokareva, S.; Zlotnik, A.; Gyrya, V. Applied Mathematical Modelling, Vol. 125 https://doi.org/10.1016/j.apm.2023.09.017	journal	January 2024
Oscillation mitigation of hyperbolicity-preserving intrusive uncertainty quantification methods for systems of conservation laws Kusch, Jonas; Schlachter, Louisa Journal of Computational and Applied Mathematics, Vol. 400 https://doi.org/10.1016/j.cam.2021.113714	journal	January 2022
A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach Heister, Timo; Wheeler, Mary F.; Wick, Thomas Computer Methods in Applied Mechanics and Engineering, Vol. 290 https://doi.org/10.1016/j.cma.2015.03.009	journal	June 2015
Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model Lee, Sanghyun; Wheeler, Mary F.; Wick, Thomas Computer Methods in Applied Mechanics and Engineering, Vol. 305 https://doi.org/10.1016/j.cma.2016.02.037	journal	June 2016
Finite-volume WENO schemes for three-dimensional conservation laws Titarev, V. A.; Toro, E. F. Journal of Computational Physics, Vol. 201, Issue 1 https://doi.org/10.1016/j.jcp.2004.05.015	journal	November 2004
Weighted essentially non-oscillatory stochastic Galerkin approximation for hyperbolic conservation laws Schlachter, Louisa; Schneider, Florian; Kolb, Oliver Journal of Computational Physics, Vol. 419 https://doi.org/10.1016/j.jcp.2020.109663	journal	October 2020
Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data Mishra, S.; Schwab, Ch. Mathematics of Computation, Vol. 81, Issue 280 https://doi.org/10.1090/S0025-5718-2012-02574-9	journal	April 2012
Multiresolution analysis for stochastic hyperbolic conservation laws Herty, M.; Kolb, A.; Müller, S. IMA Journal of Numerical Analysis https://doi.org/10.1093/imanum/drad010	journal	March 2023
Adaptive hp-Refinement for 2-D Maxwell Eigenvalue Problems: Method and Benchmarks Harmon, Jake J.; Notaros, Branislav M. IEEE Transactions on Antennas and Propagation, Vol. 70, Issue 6 https://doi.org/10.1109/TAP.2022.3145473	journal	June 2022
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/070680540	journal	January 2008
Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations Ketcheson, David I. SIAM Journal on Scientific Computing, Vol. 30, Issue 4 https://doi.org/10.1137/07070485X	journal	January 2008
A Convergent Adaptive Algorithm for Poisson’s Equation Dörfler, Willy SIAM Journal on Numerical Analysis, Vol. 33, Issue 3 https://doi.org/10.1137/0733054	journal	June 1996
Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems I: Computational Methods, A Posteriori Analysis, and Adaptive Error Control Estep, D.; MÅlqvist, A.; Tavener, S. SIAM Journal on Scientific Computing, Vol. 31, Issue 4 https://doi.org/10.1137/080731670	journal	January 2009
Error Equidistribution and Mesh Adaptation Chen, Ke SIAM Journal on Scientific Computing, Vol. 15, Issue 4 https://doi.org/10.1137/0915050	journal	July 1994
A Computational Measure Theoretic Approach to Inverse Sensitivity Problems II: A Posteriori Error Analysis Butler, T.; Estep, D.; Sandelin, J. SIAM Journal on Numerical Analysis, Vol. 50, Issue 1 https://doi.org/10.1137/100785958	journal	January 2012
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Review, Vol. 52, Issue 2 https://doi.org/10.1137/100786356	journal	January 2010
Numerical Solution of Scalar Conservation Laws with Random Flux Functions Mishra, Siddhartha; Risebro, Nils Henrik; Schwab, Christoph SIAM/ASA Journal on Uncertainty Quantification, Vol. 4, Issue 1 https://doi.org/10.1137/120896967	journal	January 2016
Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients Barajas-Solano, David A.; Tartakovsky, Daniel M. SIAM/ASA Journal on Uncertainty Quantification, Vol. 4, Issue 1 https://doi.org/10.1137/130930108	journal	January 2016
Multilevel Monte Carlo Approximation of Distribution Functions and Densities Giles, Michael B.; Nagapetyan, Tigran; Ritter, Klaus SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, Issue 1 https://doi.org/10.1137/140960086	journal	January 2015
Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation Narayan, Akil; Jakeman, John D. SIAM Journal on Scientific Computing, Vol. 36, Issue 6 https://doi.org/10.1137/140966368	journal	January 2014
A Multilevel Stochastic Collocation Method for Partial Differential Equations with Random Input Data Teckentrup, A. L.; Jantsch, P.; Webster, C. G. SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, Issue 1 https://doi.org/10.1137/140969002	journal	January 2015
Anisotropic Mesh Adaptation for Crack Detection In Brittle Materials Artina, Marco; Fornasier, Massimo; Micheletti, Stefano SIAM Journal on Scientific Computing, Vol. 37, Issue 4 https://doi.org/10.1137/140970495	journal	January 2015
A Multilevel Monte Carlo Method for Computing Failure Probabilities Elfverson, Daniel; Hellman, Fredrik; Målqvist, Axel SIAM/ASA Journal on Uncertainty Quantification, Vol. 4, Issue 1 https://doi.org/10.1137/140984294	journal	January 2016
An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data Eigel, Martin; Merdon, Christian; Neumann, Johannes SIAM/ASA Journal on Uncertainty Quantification, Vol. 4, Issue 1 https://doi.org/10.1137/15M1016448	journal	January 2016
Comparison of Adaptive Multiresolution and Adaptive Mesh Refinement Applied to Simulations of the Compressible Euler Equations Deiterding, Ralf; Domingues, Margarete O.; Gomes, Sônia M. SIAM Journal on Scientific Computing, Vol. 38, Issue 5 https://doi.org/10.1137/15M1026043	journal	January 2016
Multilevel Monte Carlo Approximation of Functions Krumscheid, S.; Nobile, F. SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, Issue 3 https://doi.org/10.1137/17M1135566	journal	January 2018
Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method Van Barel, Andreas; Vandewalle, Stefan SIAM/ASA Journal on Uncertainty Quantification, Vol. 7, Issue 1 https://doi.org/10.1137/17M1155892	journal	January 2019
A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential Jahnke, Tobias; Stein, Benny SIAM/ASA Journal on Uncertainty Quantification, Vol. 10, Issue 4 https://doi.org/10.1137/21M1440517	journal	December 2022
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
Algorithms and data structures for massively parallel generic adaptive finite element codes Bangerth, Wolfgang; Burstedde, Carsten; Heister, Timo ACM Transactions on Mathematical Software, Vol. 38, Issue 2 https://doi.org/10.1145/2049673.2049678	journal	December 2011
Multilevel Monte Carlo Path Simulation Giles, Michael B. Operations Research, Vol. 56, Issue 3 https://doi.org/10.1287/opre.1070.0496	journal	June 2008

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Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

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