A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients

Abstract

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form ut+α(t)(p1(u))x+β(t)(p2(u))xx+γ(t)(p3(u))xxx+η(t)(p4(u))xxxx=0. A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recovered the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

Numerical Study of Entropy Generation with Magnetohydrodynamics on Natural Convection in a Porous Medium using Lattice Boltzmann Method

“…≤ Gr ≤ 10 6 ). Figure (3)(4)(5)(6)(7)(8)(9) shows the results of the effects of the of dimensionless variables on the fluid flow.…”

“…The authors present a comprehensive analysis of the numerical results, including flow patterns and thermal distribution for different values of magnetic field strength, porosity, and Rayleigh number. Yang et al [3]. proposed a unified LBM for solving partial differential equations.…”