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Related Concept Videos

Carrier Transport01:21

Carrier Transport

The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.
Drift Current:
The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (vd) given by:
Lattice Energies of Ionic Crystals01:27

Lattice Energies of Ionic Crystals

Lattice energy represents the energy released when gaseous cations and anions combine to form an ionic solid, reflecting the strength of electrostatic interactions within the crystal. This process is fundamentally governed by Coulombic attraction between oppositely charged ions, where the potential energy varies inversely with the interionic distance and directly with the product of ionic charges. As ions approach one another, the electrostatic energy becomes increasingly negative, indicating a...
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
Steady, Laminar Flow Between Parallel Plates01:17

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
Maxwell-Boltzmann Distribution: Problem Solving01:20

Maxwell-Boltzmann Distribution: Problem Solving

Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
Debye–Huckel–Onsager Conductance Equation01:28

Debye–Huckel–Onsager Conductance Equation

The Debye-Hückel-Onsager equation is a cornerstone of physical chemistry, providing a method to determine the molar conductance (Λm) and molar conductance at infinite dilution (Λ°m) for uni-univalent electrolytes.Uni-univalent electrolytes are electrolytes that dissociate in solution to produce one cation with a +1 charge and one anion with a –1 charge per formula unit.This equation addresses two crucial phenomena: the asymmetry effect and the electrophoretic effect. According to this equation,...

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Related Experiment Video

Updated: Jun 25, 2026

Evolution of Staircase Structures in Diffusive Convection
07:28

Evolution of Staircase Structures in Diffusive Convection

Published on: September 5, 2018

Lattice Boltzmann model for nonlinear convection-diffusion equations.

Baochang Shi1, Zhaoli Guo

  • 1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China. sbchust@yahoo.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2009
PubMed
Summary

A novel lattice Boltzmann model accurately simulates complex nonlinear equations. This method offers a unified approach for diverse scientific models, showing excellent agreement with existing solutions.

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Area of Science:

  • Computational physics
  • Numerical analysis
  • Nonlinear dynamics

Background:

  • Convection-diffusion equations are fundamental in modeling various physical phenomena.
  • Existing numerical methods can struggle with nonlinear terms and complex-valued equations.
  • A unified and efficient modeling approach is needed for diverse nonlinear evolutionary equations.

Purpose of the Study:

  • To propose a versatile lattice Boltzmann model for convection-diffusion equations.
  • To demonstrate the model's applicability to a range of real and complex-valued nonlinear equations.
  • To validate the model's accuracy and efficiency through detailed simulations.

Main Methods:

  • Development of a lattice Boltzmann model by selecting an appropriate equilibrium distribution function.
  • Adaptation of the model using real/complex-valued distribution functions and relaxation times.
  • Implementation of detailed numerical simulations for various nonlinear equations.

Main Results:

  • The proposed lattice Boltzmann model successfully simulates nonlinear convection-diffusion equations.
  • The model is effective for diverse equations including nonlinear Schrödinger, Ginzburg-Landau, Burgers-Fisher, nonlinear heat conduction, and sine-Gordon equations.
  • Numerical results show excellent agreement with analytical solutions and previously reported numerical findings.

Conclusions:

  • The developed lattice Boltzmann model provides a robust and unified numerical framework.
  • This approach offers a powerful tool for studying complex nonlinear phenomena across different scientific domains.
  • The model's accuracy and versatility are confirmed by extensive simulations.