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Entropic lattice Boltzmann model for Burgers's equation
Bruce M Boghosian1, Peter Love, Jeffrey Yepez
1Department of Mathematics, Tufts University, Medford, MA 02155, USA. bruce.boghosian@tufts.edu
Summary
Entropic lattice Boltzmann models offer stable numerical simulations for hydrodynamic equations. However, these models can introduce oscillations near shocks at very low viscosity, potentially limiting their accuracy for turbulent flow simulations.
Area of Science:
- Computational Fluid Dynamics
- Nonlinear Dynamics
- Numerical Analysis
Background:
- Entropic lattice Boltzmann models are discrete-velocity hydrodynamic models with Lyapunov functions, ensuring nonlinear stability.
- These models have been successfully applied to Navier-Stokes equations and proposed as subgrid models for turbulence.
- Their inherent stability makes them attractive for numerically integrating complex hydrodynamic equations.
Purpose of the Study:
- To develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension.
- To evaluate the model's effectiveness and stability in simulating Burgers's equation.
- To investigate potential limitations of entropic lattice Boltzmann models in scenarios with small gradient scale lengths.
Main Methods:
- Development of a novel entropic lattice Boltzmann model specifically for one-dimensional Burgers's equation.
- Numerical simulation of Burgers's equation across a range of viscosity values.
- Analysis of model behavior, focusing on stability and accuracy, particularly near shock regions.
Main Results:
- The entropic lattice Boltzmann model effectively simulates Burgers's equation, showing no instability at moderate to high viscosity.
- At very low viscosity, bounded amplitude oscillations emerge near shocks where gradient scale lengths approach grid size.
- The amplitude of these oscillations increases as viscosity decreases, indicating a potential loss of accuracy.
Conclusions:
- Entropic lattice Boltzmann models provide a stable and effective method for simulating Burgers's equation.
- Model accuracy can be compromised at very low viscosity due to grid-scale shock phenomena.
- These findings suggest that the entropic lattice Boltzmann approach may have limitations as a subgrid turbulence model for Navier-Stokes equations when resolving small-scale features.
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