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Discretization of the Burgers' equation as a port-Hamiltonian system

Lorenzo Agostini, Michel Fournié, Ghislain Haine

Abstract

The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.

Discretization of the Burgers' equation as a port-Hamiltonian system

Abstract

The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work, port-Hamiltonian formulations for both the inviscid and the viscous Burgers' equations are proposed, enabling a representation that incorporates both convective and dissipative effects. Boundary control and observation are naturally handled within this framework. Applying a dedicated finite element method, a finite-dimensional port-Hamiltonian system is derived. The relationship between time step, spatial resolution, and viscosity required to achieve numerical stability is analyzed. Numerical experiments validate the effectiveness of the approach.
Paper Structure (12 sections, 8 theorems, 49 equations, 4 figures, 1 table)

This paper contains 12 sections, 8 theorems, 49 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inviscid model
  3. Power balance
  4. A change of Hamiltonian
  5. Port-Hamiltonian representation
  6. Structure-preserving discretization
  7. Viscous model
  8. Port-Hamiltonian representation
  9. Structure-preserving discretization
  10. Numerical simulations
  11. Stability
  12. Conclusion

Key Result

Lemma 1

For smooth solutions, the kinetic energy satisfies the power balance After the formation of a shock, located at $x = x_s(t)$, and assuming the left limits and the right limits the time derivative of the kinetic energy becomes In other words, a dissipation occurs at shock.

Figures (4)

  • Figure 1: Velocity profiles of the inviscid case
  • Figure 2: Velocity profiles of the viscous case
  • Figure 3: Hamiltonian and dissipation evolution of the viscous case
  • Figure 4: Kinetic energy and dissipation evolution of the viscous case

Theorems & Definitions (16)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 4
  • Remark 5
  • Proposition 5
  • ...and 6 more