A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems

Elton, A B.H.

doi:10.2172/6480937

Title: A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems

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Abstract

A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.

Authors:: Elton, A B.H.

Publication Date:: Mon Sep 24 00:00:00 EDT 1990

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: DOE/DP

OSTI Identifier:: 6480937

Report Number(s):: UCRL-LR-105090
ON: DE91002566

DOE Contract Number:: W-7405-ENG-48

Resource Type:: Technical Report

Resource Relation:: Other Information: Thesis (Ph.D.)

Country of Publication:: United States

Language:: English

Subject:: 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ADVECTION; NONLINEAR PROBLEMS; DIFFUSION; BOLTZMANN STATISTICS; CHAPMAN-ENSKOG THEORY; CONVERGENCE; EIGENVALUES; EIGENVECTORS; FINITE DIFFERENCE METHOD; NAVIER-STOKES EQUATIONS; NUMERICAL ANALYSIS; DIFFERENTIAL EQUATIONS; EQUATIONS; ITERATIVE METHODS; MASS TRANSFER; MATHEMATICS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; 656001* - Condensed Matter Physics- Solid-State Plasma; 990200 - Mathematics & Computers; 640400 - Fluid Physics

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                    Elton, A B.H. A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems.  United States: N. p., 1990. 
        Web.  doi:10.2172/6480937.

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                    Elton, A B.H. A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems.  United States.  https://doi.org/10.2172/6480937

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                    Elton, A B.H. 1990.  
        "A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems".  United States.  https://doi.org/10.2172/6480937.  https://www.osti.gov/servlets/purl/6480937.

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@article{osti_6480937,

  title        = {A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems},

  author       = {Elton, A B.H.},

  abstractNote = {A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.},

  doi          = {10.2172/6480937},

  url          = {https://www.osti.gov/biblio/6480937},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Mon Sep 24 00:00:00 EDT 1990},

  month        = {Mon Sep 24 00:00:00 EDT 1990}

}

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