Research Projects

Structure Preserving Approximations for Robust Computation of Conservation Laws and Related Equations

Principal investigator

Prof. Dr. Siddhartha Mishra, Seminar for Applied Mathematics, ETH Zurich

Researchers

MSc. Filippo Leonardi, Seminar for Applied Mathematics, ETH Zurich
MSc. Kjetil Lye, Seminar for Applied Mathematics, ETH Zurich

Start date: 01.12.2012 / End date: 30.11.2017

Description

Project No. 306279, Acronym SPARCCLE
Many interesting systems in physics and engineering are mathematically modeled by first-order non-linear hyperbolic partial differential equations termed as systems of conservation laws. Examples include the Euler equations of aerodynamics, the shallow water equations of oceanography, multi-phase flows in a porous medium (uesed in the oil industry), equations of non-linear elasticity and the MHD equations of plasma physics. Numerical methods are the key tools to study these equations and to simulate interesting phenomena such as shock waves. Despite the intense development of numerical methods for the past three decades and great success in applying these methods to large scale complex physical and engineering simulations, the massive increase in computational power in recent years has exposed the inability of state of the art schemes to simulate very large, multiscale, multiphysics three dimensional problems on complex geometries. In particular, problems with strong shocks that depend explicitly on underlying small scale effects, involve geometric constraints like vorticity and require uncertain inputs such as random initial data and source terms, are beyond the range of existing methods. The main goal of this project will be to design space-time adaptive \emph{structure preserving} arbitrarily high-order finite volume and discontinuous Galerkin schemes that incorporate correct small scale information and provide for efficient uncertainty quantification. These schemes will tackle emerging grand challenges and dramatically increase the range and scope of numerical simulations for systems modeled by hyperbolic PDEs. Moreover, the schemes will be implemented to ensure optimal performance on emerging massively parallel hardware architecture. The resulting publicly available code can be used by scientists and engineers to study complex systems and design new technologies.

Publications

Submitted Publications

K. Lye. Multilevel Monte-Carlo for measure valued solutions, in review (2016), SAM Report 2016-51
S. M. May. Spacetime discontinuous Galerkin methods for solving convection-diffusion systems, in review (2015), SAM Report 2015-05

Refereed Publications in Journals or Collections

U. S. Fjordholm, K. Lye and S. Mishra. Numerical approximation of statistical solutions of scalar conservation laws, SIAM J. Num. Anal., 56/5 (2018), pp. 2989-3009, SAM Report 2017-52 , doi
A. Hiltebrand, S. Mishra and C. Parés. Entropy-stable space-time DG schemes for non-conservative hyperbolic systems, ESAIM: M2AN, 52/3 (2018), pp. 995-1022, SAM Report 2017-39 , doi
F. Leonardi. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations, Discrete and Continuous Dynamical Systems - Series S, 11/5 (2018), pp. 941-961, SAM Report 2017-06 , doi
U. Fjordholm, S. Lanthaler and S. Mishra. Statistical solutions of hyperbolic conservation laws: Foundations, Arch. Rat. Mech. Anal., 226/2 (2017), pp. 809-849, SAM Report 2016-59 , doi
U. Fjordholm, R. Kaeppeli, S. Mishra and E. Tadmor. Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math., 17/3 (2017), pp. 763-827, SAM Report 2014-33 , doi
A. Perego, R. Cabezón and R. Käppeli. An advanced leakage scheme for neutrino treatment in astrophysical simulations, The Astrophysical Journal Supplement Series, 223/22 (2016), SAM Report 2015-41 , doi
R. Käppeli and S. Mishra. A well-balanced finite volume scheme for the Euler equations with gravitation, Astronomy and Astrophysics, 587/A94 (2016), SAM Report 2015-40 , doi
F. Leonardi, S. Mishra and Ch. Schwab. Numerical approximation of statistical solutions of planar, incompressible flows, Math. Models and Method in Applied Sciences, 26/13 (2016), pp. 2471-2524, SAM Report 2015-27 , doi
A. Hiltebrand and S. Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography, Networks and Heterogeneous Media, 11/1 (2016), pp. 145-162, SAM Report 2015-13 , doi
S. Lanthaler and S. Mishra. Computation of measure-valued solutions for the incompressible Euler equations, Math. Mod. Meth. Appl. Sci. (M3AS), 25/11 (2015), pp. 2043-2088, SAM Report 2014-34 , doi
C. Sanchez-Linares, M. de la Asuncion, M. Castro, S. Mishra and J. Šukys. Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis, Appl. Math. Modelling, 39/23-24 (2015), pp. 7211-7226, SAM Report 2014-24 , doi
A. Hiltebrand and S. Mishra. Efficient preconditioners for a shock capturing space-time discontinuous Galerkin method for systems of conservation laws, Commun. Comput. Phys., 17/5 (2015), pp. 1320-1359, SAM Report 2014-04 , doi
J. Ernest, P. LeFloch and S. Mishra. Schemes with Well-Controlled Dissipation, SIAM J. Numerical Analysis, 53/1 (2015), pp. 674-699, SAM Report 2013-41 , doi
G. M. Coclite, S. Mishra, N. Risebro and F. Weber. Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media, Comp. GeoSci., 18/5 (2014), pp. 637-659, SAM Report 2013-44 , doi
R. Käppeli and S. Mishra. Well-balanced schemes for the Euler equations with gravitation, Journal of Computational Physics, 259 (2014), pp. 199-219, SAM Report 2013-05 , doi
A. Hiltebrand and S. Mishra. Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numerische Mathematik, 126/1 (2014), pp. 103-151, SAM Report 2012-07 , doi
P. LeFloch and S. Mishra. Numerical methods with controlled dissipation for small-scale dependent shocks, Acta Numerica, 23 (2014), pp. 743-816, doi
I. Averbukh, D. Ben-Zvi, S. Mishra and N. Barkai. Scaling morphogen gradients during tissue growth by a cell division rule, Development, 141 (2014), pp. 2150-2156, doi
G. M. Coclite, L. Di Ruvo, J. Ernest and S. Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes, Netw. Heterog. Media, 8/4 (2013), pp. 969-984, SAM Report 2012-30

Books and Book Chapters

A. Hiltebrand and S. Mishra. Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method, EMS Series of Congress Reports, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis : The Helge Holden Anniversary Volume (2018), pp. 287-318, SAM Report 2014-17 , doi
S. Mishra and Ch. Schwab. Monte-Carlo Finite-Volume Methods in Uncertainty Quantification for Hyperbolic Conservation Laws, Uncertainty Quantification for Hyperbolic and Kinetic Equations, SEMA SIMAI, volume 14 (2017), pp. 231-277, SAM Report 2017-38 , doi
R. Abgrall and S. Mishra. Uncertainty quantification for hyperbolic systems of conservation laws, Handbook of numerical methods for hyperbolic problems, 18 (2017), pp. 507-544, SAM Report 2016-58 , doi
S. Mishra. Numerical methods for conservation laws with discontinuous coefficients, Handbook of numerical methods for hyperbolic problems, 18 (2017), pp. 479-506, SAM Report 2016-57 , doi

Other Publications

S. M. May. Spacetime discontinuous Galerkin methods for convection-diffusion equations, Proc. Bull. Braz. Math. Soc., New Series, 47/2 (2016), pp. 561-573, doi
R. Käppeli and S. Mishra. Structure preserving schemes, Proc. ASTRONUM 2013, 488 (2014), pp. 231, SAM Report 2014-02 , doi
F. Fuchs, A. McMurry, S. Mishra and N.H. Risebro. Explicit and implicit finite volume schemes for radiation on wave propagation in stratified atmospheres, Proc. of the Hyperbolic Problems 2012, AIMS Series on Applied Mathematics, 8 (2014), pp. 41-59, SAM Report 2013-40
F. Weber. Robust finite difference schemes for a nonlinear variational wave equation modeling liquid crystals (2013), SAM Report 2013-43

Theses

A. Hiltebrand. Entropy-stable discontinuous Galerkin finite element methods with streamline diffusion and shock-capturing for hyperbolic systems of conservation laws (2014), PhD Thesis, ETH Diss Nr. 22279, Examiner Prof. Dr. Siddhartha Mishra, doi

Fundings

Seventh Framework Programme (FP7)
European Research Council - ERC

Contact

Siddhartha Mishra