Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conformal variational discretisation of infinite dimensional Hamiltonian systems with gradient flow dissipation

Damiano Lombardi, Cecilia Pagliantini

TL;DR

This work presents a conformal variational discretisation for infinite-dimensional Hamiltonian systems with gradient-flow dissipation, by first deriving a mixed variational formulation that separates conservative and dissipative effects and then applying a Galerkin spatial discretisation. The semi-discrete scheme retains a finite-dimensional Poisson structure and a symmetric dissipative part, preserves invariants via discrete Casimirs, and converges to the continuous mixed formulation under standard Lipschitz assumptions. Convergence results are complemented by temporal discretisations (AVF and implicit midpoint) that respect Hamiltonian and dissipation properties in practice. Numerical tests on the Korteweg–de Vries equation and 2D Navier–Stokes equations on the torus and sphere validate the theory, showing accurate invariants preservation, correct dissipation, and robust mesh convergence.

Abstract

Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a nonconservative part, in the form of gradient flow dissipation. Traditional numerical approximations of this class of problem typically fail to retain the separation into conservative and nonconservative parts hence leading to unphysical solutions. In this work we propose a mixed variational method that gives a semi-discrete problem with the same geometric structure as the infinite-dimensional problem. As a consequence the conservation laws and the dissipative terms are retained. A priori convergence estimates on the solution are established. Numerical tests of the Korteweg-de Vries equation and of the two-dimensional Navier-Stokes equations on the torus and on the sphere are presented to corroborate the theoretical findings.

Conformal variational discretisation of infinite dimensional Hamiltonian systems with gradient flow dissipation

TL;DR

This work presents a conformal variational discretisation for infinite-dimensional Hamiltonian systems with gradient-flow dissipation, by first deriving a mixed variational formulation that separates conservative and dissipative effects and then applying a Galerkin spatial discretisation. The semi-discrete scheme retains a finite-dimensional Poisson structure and a symmetric dissipative part, preserves invariants via discrete Casimirs, and converges to the continuous mixed formulation under standard Lipschitz assumptions. Convergence results are complemented by temporal discretisations (AVF and implicit midpoint) that respect Hamiltonian and dissipation properties in practice. Numerical tests on the Korteweg–de Vries equation and 2D Navier–Stokes equations on the torus and sphere validate the theory, showing accurate invariants preservation, correct dissipation, and robust mesh convergence.

Abstract

Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a nonconservative part, in the form of gradient flow dissipation. Traditional numerical approximations of this class of problem typically fail to retain the separation into conservative and nonconservative parts hence leading to unphysical solutions. In this work we propose a mixed variational method that gives a semi-discrete problem with the same geometric structure as the infinite-dimensional problem. As a consequence the conservation laws and the dissipative terms are retained. A priori convergence estimates on the solution are established. Numerical tests of the Korteweg-de Vries equation and of the two-dimensional Navier-Stokes equations on the torus and on the sphere are presented to corroborate the theoretical findings.

Paper Structure

This paper contains 20 sections, 14 theorems, 143 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian systems with gradient flow dissipation
  3. Infinite dimensional Hamiltonian dynamics
  4. Gradient flow dissipation
  5. Hamiltonian systems with gradient flow dissipation
  6. The case of metriplectic or GENERIC systems
  7. The variational formulation of the problem
  8. Mixed formulation
  9. Equivalence of the variational and mixed formulations
  10. The semi-discrete problem
  11. Conformal variational discretisation
  12. Geometric structure of the semi-discrete problem
  13. Conservation properties of the semi-discrete problem
  14. The case of metriplectic or GENERIC systems
  15. Convergence results
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let $\xi\in X_{-s+d}$ and $u\in K_{\mathcal{J}}$. If $\left<{\xi, v}\right>_0 = 0$ for all $v\in X_s$, then

Figures (10)

  • Figure 1: On the left: zoom on the final time numerical solution compared to the analytical one. On the right, mesh convergence in log-log scale.
  • Figure 2: Mass (on the left) and Hamiltonian (on the right) conservation in time, for the solution of the KdV equation with no dissipation.
  • Figure 3: On the left, time evolution of the solution of the dissipative KdV equation; the red curve is the initial condition evaluated at the mesh nodes. On the right, the discrete residual in time of the entropy production rate.
  • Figure 4: Initial condition for the stream function $\Psi$ (on the left), and convergence in mesh (on the right), for the Navier-Stokes equation solution on the torus, when $\nu=10^{-2}$.
  • Figure 5: Hamiltonian decay (left), and error in semi-logartithmic scale (right).
  • ...and 5 more figures

Theorems & Definitions (36)

  • Definition 2.1: Hilbert scale
  • Definition 2.2: Morphism of order $d$
  • Definition 2.3: Poisson bracket
  • Definition 2.4: Invariants of motion
  • Definition 2.5: Casimir invariants
  • Example 2.1: Heat equation
  • Example 2.2: Double bracket dissipation
  • Example 2.3: Advection-diffusion equation
  • Example 2.4: Isentropic KdV equation
  • Lemma 3.1
  • ...and 26 more