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Main topics of the NumHyp-2015' discussion session

Denys Dutykh, Laurent Gosse

Abstract

Three main topics were raised in this discussion session, which took place on the 19th of June at the NumHyp-2015 meeting: nonlinear resonance for 1D systems of balance laws, dispersive extensions of standard hyperbolic conservation laws, and the validation of weakly dispersive shallow water wave models. An introductory overview with many bibliographic references is provided for all these topics. Based on kinetic formulation, a numerical strategy that can overcome resonance issues is presented, and a well-balanced (WB) technique for Vlasov-Fokker-Planck equations is outlined. This WB scheme relies on the spectral representation of stationary solutions.

Main topics of the NumHyp-2015' discussion session

Abstract

Three main topics were raised in this discussion session, which took place on the 19th of June at the NumHyp-2015 meeting: nonlinear resonance for 1D systems of balance laws, dispersive extensions of standard hyperbolic conservation laws, and the validation of weakly dispersive shallow water wave models. An introductory overview with many bibliographic references is provided for all these topics. Based on kinetic formulation, a numerical strategy that can overcome resonance issues is presented, and a well-balanced (WB) technique for Vlasov-Fokker-Planck equations is outlined. This WB scheme relies on the spectral representation of stationary solutions.
Paper Structure (16 sections, 41 equations, 4 figures, 1 table)

This paper contains 16 sections, 41 equations, 4 figures, 1 table.

Table of Contents

  1. Nonlinear resonance: issues and advances
  2. Non-interacting asymptotic states and Temple system
  3. Non-strict hyperbolicity and wave trapping effects
  4. A numerical case-study for isentropic Euler with $\gamma = 3$
  5. A fictitious (but resonant) model of 1D semiconductor
  6. Disperse flows and hyperbolic-kinetic couplings
  7. Sediment transport in shallow water models
  8. A simple Burgers/Vlasov--Fokker--Planck toy model
  9. Dispersive wave modelling
  10. Water wave propblem
  11. The method of holomorphic variables
  12. Computation of steady fully nonlinear solutions
  13. Extended fully nonlinear shallow water equations
  14. Linear approximation
  15. Validation on solitary waves
  16. ...and 1 more sections

Figures (4)

  • Figure 4: Conformal map of the physical domain into a uniform strip.
  • Figure 5: Iso-horizontal (left) and iso-vertical (right) velocities under a large solitary wave. Lines correspond to the iso-values computed in the 'fixed' Frame of reference where the the fluid is at rest in the far field $x\to\pm\infty$. Taken from Dutykh2013b.
  • Figure 6: Speed--amplitude relations for solitary waves in SGN, eSGN and the full Euler equations ($\alpha = 6/5$).
  • Figure 7: Comparison of the runup reduction effect for various ad-hoc friction terms and the random bottom perturbation model. The horizontal axis represents the friction coefficient $c_f$ for deterministic computations and $\sigma$ for the random roughness model (blue solid line).