Lattice Boltzmann model for high-order nonlinear partial differential equations

Area of Science:

• Computational physics
• Applied mathematics
• Numerical analysis

Background:

• High-order nonlinear partial differential equations (PDEs) are crucial in modeling complex physical phenomena.
• Existing lattice Boltzmann (LB) models face challenges in accurately solving these higher-order PDEs.
• Specific examples include the (m)KdV, KdV-Burgers, and Kuramoto-Sivashinsky equations.

Purpose of the Study:

• To propose a generalized lattice Boltzmann (LB) model for a broad class of high-order nonlinear PDEs.
• To demonstrate the model's capability in solving diverse and complex differential equations.
• To enhance the accuracy and applicability of LB methods for nonlinear physics.

Main Methods:

• Development of a general lattice Boltzmann (LB) model for PDEs of the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0.
• Introduction of specific auxiliary moments to achieve correct equilibrium distribution function moments.
• Rigorous Chapman-Enskog analysis to recover the target high-order nonlinear PDE.
• Extensive numerical simulations to validate the model's performance.

Main Results:

• The proposed LB model successfully solves various high-order nonlinear PDEs, including special cases like the (m)KdV and Kawahara equations.
• Chapman-Enskog analysis confirms the accurate recovery of the governing high-order nonlinear PDE.
• Numerical results show excellent agreement with analytical solutions.
• The new LB model demonstrates superior accuracy compared to existing methods for these equations.

Conclusions:

• The generalized LB model provides an accurate and efficient numerical tool for high-order nonlinear PDEs.
• The introduction of auxiliary moments is key to the model's improved performance.
• This work advances the application of LB methods in computational physics and applied mathematics.