Stabilization and Limiting Techniques for Galerkin Approximations of Hyperbolic Conservation Laws With High Order Finite Elements

• Applicant: Professor Dr. Dmitri Kuzmin
• Subject Area: Mathematics
• Term: from 2017 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 387630025

• Final Report Year: 2024

Final Report Abstract

The ambition of this research project was to develop a new generation of physics-compatible high-resolution finite element schemes for nonlinear hyperbolic problems. In this context, failure to suppress spurious oscillations and satisfy appropriate constraints may result in numerical instabilities and nonphysical artifacts. The framework of monolithic convex limiting (MCL) developed in this project makes it possible to ensure preservation of invariant domains (i.e., positivity preservation for quasi-concave functions of conserved variables) and validity of local discrete maximum principles for scalar quantities of interest. Limiter-based algebraic fixes were designed for enforcing entropy stability conditions in semi-discrete and fully discrete schemes. Property-preserving time discretization limiters were developed for intermediate and final stages of high-order Runge-Kutta methods. The difficult task of achieving optimal accuracy and performance with continuous and discontinuous Galerkin discretizations of arbitrary high order was accomplished for nodal Bernstein and Legendre-Gauss-Lobatto (LGL) bases using localization to subcells and sparsification of discrete operators. Weighted essentially nonoscillatory (WENO) reconstructions were used to construct novel smoothness sensors for nonlinear stabilization terms. Very significant advances have also been made in the theoretical analysis of algebraic flux correction schemes. New a priori error estimates were derived for linear transport equations. The ability of each limiting technique to enforce the desired properties is guaranteed by a rigorous proof. The entropy stability of MCL-type finite element discretizations was exploited in proofs of Lax-Wendroff consistency and convergence to dissipative weak solutions of the compressible Euler equations. The main outcomes of the project were summarized and many new results were included in the book "Property-Preserving Numerical Schemes for Conservation Laws" (470 pages + supplementary materials) published by World Scientific in 2023.

Publications

• A partition of unity approach to adaptivity and limiting in continuous finite element methods. Computers & Mathematics with Applications, 78(3), 944-957.
  Kuzmin, Dmitri; Quezada de Luna, Manuel & Kees, Christopher E.
  
• Physics-Compatible Finite Element Methods for Scalar and Tensorial Advection Problems. Springer Fachmedien Wiesbaden.
  Lohmann, Christoph
  
• Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 372, 113370.
  Kuzmin, Dmitri & Quezada de Luna, Manuel
  
• Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Computers & Fluids, 213, 104742.
  Kuzmin, Dmitri & Quezada de Luna, Manuel
  
• Gradient-Based Limiting and Stabilization of Continuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, 331-339. Springer International Publishing.
  Kuzmin, Dmitri
  
• Locally bound-preserving enriched Galerkin methods for the linear advection equation. Computers & Fluids, 205, 104525.
  Kuzmin, Dmitri; Hajduk, Hennes & Rupp, Andreas
  
• Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. Computer Methods in Applied Mechanics and Engineering, 359, 112658.
  Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio & Abgrall, Remi
  
• Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting. Computers & Fluids, 200, 104451.
  Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; Tomov, Vladimir; Tomas, Ignacio & Shadid, John N.
  
• Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 361, 112804.
  Kuzmin, Dmitri
  
• Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws. Journal of Computational Physics, 411, 109411.
  Kuzmin, Dmitri & Quezada de Luna, Manuel
  
• A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 373, 113569.
  Kuzmin, Dmitri
  
• An algebraic flux correction scheme facilitating the use of Newton-like solution strategies. Computers & Mathematics with Applications, 84, 56-76.
  Lohmann, Christoph
  
• Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems. Journal of Numerical Mathematics, 29(4), 307-322.
  Kuzmin, Dmitri
  
• Algebraically Constrained Finite Element Methods for Hyperbolic Problems With Applications to Geophysics and Gas Dynamics. Ph.D. thesis, TU Dortmund University
  Hajduk, Hennes
  
• Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws. Journal of Scientific Computing, 91(1).
  Kuzmin, Dmitri; Quezada de Luna, Manuel; Ketcheson, David I. & Grüll, Johanna
  
• Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems. Computer Methods in Applied Mechanics and Engineering, 389, 114428.
  Kuzmin, Dmitri; Hajduk, Hennes & Rupp, Andreas
  
• Analysis of algebraic flux correction schemes for semi-discrete advection problems. BIT Numerical Mathematics, 63(1).
  Hajduk, Hennes & Rupp, Andreas
  
• Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws. Journal of Computational Physics, 487, 112153.
  Kuzmin, Dmitri & Vedral, Joshua
  
• Property-Preserving Numerical Schemes for Conservation Laws. WORLD SCIENTIFIC.
  Kuzmin, Dmitri & Hajduk, Hennes
  
• Dissipative WENO stabilization of high-order discontinuous Galerkin methods for hyperbolic problems.
  Vedral, Joshua
  
• Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods. Communications on Applied Mathematics and Computation, 6(3), 1860-1898.
  Rueda-Ramírez, Andrés M.; Bolm, Benjamin; Kuzmin, Dmitri & Gassner, Gregor J.
  
• Well-balanced convex limiting for finite element discretizations of steady convection-diffusion-reaction equations. Journal of Computational Physics, 518, 113305.
  Knobloch, Petr; Kuzmin, Dmitri & Jha, Abhinav
  
• Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations. Journal of Computational Physics, 523, 113687.
  Moujaes, Paul & Kuzmin, Dmitri
  
• Strongly consistent low-dissipation WENO schemes for finite elements. Applied Numerical Mathematics, 210, 64-81.
  Vedral, Joshua; Rupp, Andreas & Kuzmin, Dmitri