ICM 2026 Invited Section Lecture - Michael Dumbser

Value of the Information & Strength of the Argument

The value of this contribution lies in its dual approach to classical entropy-stable schemes: exchanging the roles of entropy and energy at the discrete level. This is demonstrated to be more general, handling multi-phase flows or models with multiple entropy equations (like Gavriluk’s turbulence model) where only one energy conservation law exists. The argumentation is rigorous and systematic: the speaker provides a clear mathematical framework, proves local and global discrete energy conservation by construction, and validates the schemes through numerical experiments (Sod/Lax shock tubes, Navier-Stokes limits, experimental comparisons for shallow water turbulence) showing correct shock speeds. The discussion of the discrete Schwarz theorem offers an elegant, simpler alternative to finite element exterior calculus for preserving involutions, elegantly reducing both curl-free and divergence-free conditions to a single discrete symmetry of second derivatives.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is exceptionally high, as the talk is structured around rigorous derivations and proofs, and explicitly acknowledges current limitations (no scheme yet satisfies both properties of energy conservation and involution preservation simultaneously). The speaker carefully cites foundational works by Godunov (1961), Tadmor’s pioneering entropy stable fluxes, Peshkov & Romenski’s model, and Gavriluk’s turbulence model. The title is perfectly aligned with the content, which focuses exactly on compatible numerical schemes for the specified class of PDEs. The lecture follows a logical progression from theory to discrete schemes and numerical verification.

The title perfectly mirrors the content, which is entirely devoted to presenting new compatible numerical schemes for thermodynamically compatible hyperbolic PDE systems.

9/10

The talk presents a highly rigorous mathematical exposition, underpinned by formal derivations, theorems (e.g., discrete energy conservation, entropy inequality, discrete Schwarz theorem), and numerical validation against exact solutions. It builds on established foundational work (Godunov, Tadmor, Peshkov-Romenski) and clearly delineates open problems.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction to thermodynamically compatible hyperbolic systems and the legacy of Godunov's 1961 discovery.
• 04:00Derivation of HTC systems using generating potentials, symmetric hyperbolicity, and thermodynamic dual variables.
• 08:00Presentation of the Godunov-Peshkov-Romenski (GPR) model, including equations for solids, fluids, and hyperbolic heat conduction.
• 16:00Design of a finite volume scheme that discretizes the entropy equation and proves discrete total energy conservation.
• 24:00Comparison with Hamiltonian ODE integration and explanation of why discretizing entropy is more general for multi-entropy systems.
• 32:00Introduction of the discrete Schwarz theorem approach for preserving involutions, applied to linear acoustics and Maxwell equations.

• Godunov, S. K. (1961). An interesting class of quasilinear systems. — Cited as the foundational work introducing thermodynamically compatible systems and their symmetric hyperbolic form.
• Tadmor, E. (1987). The numerical viscosity of entropy stable schemes for systems of conservation laws. — Mentioned as the source of the entropy-preserving flux used in the dual fashion to achieve energy conservation.
• Gavriluk, S. et al. Hyperbolic turbulence models. — Cited to illustrate the generality of the approach for systems with multiple entropy equations or independent stress components.
• Peshkov, I. & Romenski, E. (2016) - GPR model for continuum mechanics. — Foundation of the unified model for elastodynamics and fluid dynamics used throughout the talk.

• Godunov's scheme (1959) — Established the foundational finite volume framework and Riemann solvers whose stability properties are extended here.
• Tadmor's entropy stable schemes (1987) — The talk explicitly builds on Tadmor's entropy-preserving flux, applying it in a dual fashion to conserve energy.

• Classical conservative schemes for Euler equations (replacing entropy with energy) — The standard approach in CFD of discretizing the energy equation and discarding the entropy equation is directly contrasted by the author, who argues it is unnatural and fails for systems with multiple entropy laws.