[Paper Review] Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes

François Dubois|arXiv (Cornell University)|Jan 2, 2014

Lattice Boltzmann Simulation Studies20 references22 citations

TL;DR

This paper proposes a vectorial lattice Boltzmann scheme based on kinetic representation of the dual entropy to simulate strong nonlinear waves in hyperbolic conservation laws, specifically applying it to shallow water equations in one and two dimensions. The method successfully captures shock waves, reflections, and unstationary flows with results comparable to Godunov finite volume schemes, though with higher numerical viscosity.

ABSTRACT

We show that an hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes with the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two space dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and unstationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in unstationary situations.

Motivation & Objective

• To extend the kinetic representation of dual entropy from 1D to multidimensional systems of hyperbolic conservation laws.
• To address the challenge of simulating strong nonlinear waves, including shock formation and interaction, in systems like shallow water equations.
• To develop a stable and consistent lattice Boltzmann framework that preserves entropy structure and handles discontinuities.
• To evaluate the method's performance on benchmark test cases involving stationary and unstationary shock dynamics.

Proposed method

• The method uses N particle distributions f_j^k for N conserved variables, with discrete velocities v_j, to represent the system via moment-based equations.
• It applies the Perthame-Bouchut hypothesis, decomposing the dual entropy into convex functions associated with discrete velocity sets.
• The equilibrium distributions are constructed using the dual entropy and entropy variables, ensuring consistency with the underlying hyperbolic system.
• A BGK-type collision model is used with equal relaxation parameters s_j = 1.8 for stability, and multiple relaxation times are implemented for non-equilibrium moments.
• Boundary conditions are applied via bounce-back and anti-bounce-back schemes for wall, inflow, and outflow conditions.
• The scheme is validated against exact solutions and Godunov finite volume results on structured grids.

Experimental results

Research questions

• RQ1 Can vectorial lattice Boltzmann schemes based on dual entropy decomposition accurately simulate strong nonlinear waves in hyperbolic systems?
• RQ2 How does the method perform in capturing stationary shock reflections and unstationary shock interactions in shallow water flows?
• RQ3 What is the impact of numerical viscosity on the scheme’s accuracy compared to high-resolution finite volume methods?
• RQ4 Can the kinetic representation of dual entropy be generalized to multidimensional systems with multiple conserved variables?
• RQ5 How do mesh refinement and parameter choices affect stability and convergence in unstationary simulations?

Key findings

• The vectorial lattice Boltzmann scheme successfully captures stationary shock reflection in 2D shallow water flows, with results matching the exact solution and Godunov finite volume method on 140×80 grids.
• For the unstationary Emery test case at Froude number 3, the scheme reproduces complex shock wave interactions and nonlinear wave propagation at t=1/2 and t=4 on a 480×120 mesh.
• The scheme exhibits high numerical viscosity, resulting in less accurate unstationary results compared to the Godunov scheme, which is first-order accurate but more precise in transient regimes.
• The method maintains consistency with the underlying entropy structure, ensuring thermodynamically consistent evolution of the system.
• Mesh refinement shows convergence, with density profiles from the lattice Boltzmann and Godunov schemes aligning closely on the finest grid.
• The D2Q5Q4Q4 scheme configuration with λ=80 and s_j=1.8 ensures numerical stability, though the high λ value leads to small time steps and slow computation.