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A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations

S. V. Raghurama Rao, K. S. Shrinath, Ankit Ruhi, Veeredhi Vasudeva Rao

Abstract

A framework of finite-velocity model based Boltzmann equation has been developed for convection-diffusion equations. These velocities are kept flexible and adjusted to control numerical diffusion. A flux difference splitting based kinetic scheme is then introduced for solving a wide variety of nonlinear convection-diffusion equations numerically. Based on this framework, a generalized kinetic Lax-Wendroff scheme is also derived, recovering the classical Lax-Wendroff method as one of the choices. Further, a total variation diminishing version of this kinetic flux difference splitting scheme is presented, combining it with the kinetic Lax-Wendroff scheme using a limiter function. The numerical scheme has been extensively tested and the results for benchmark test cases, for 1D and 2D nonlinear convection and convection-diffusion equations, are presented.

A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations

Abstract

A framework of finite-velocity model based Boltzmann equation has been developed for convection-diffusion equations. These velocities are kept flexible and adjusted to control numerical diffusion. A flux difference splitting based kinetic scheme is then introduced for solving a wide variety of nonlinear convection-diffusion equations numerically. Based on this framework, a generalized kinetic Lax-Wendroff scheme is also derived, recovering the classical Lax-Wendroff method as one of the choices. Further, a total variation diminishing version of this kinetic flux difference splitting scheme is presented, combining it with the kinetic Lax-Wendroff scheme using a limiter function. The numerical scheme has been extensively tested and the results for benchmark test cases, for 1D and 2D nonlinear convection and convection-diffusion equations, are presented.
Paper Structure (33 sections, 324 equations, 23 figures, 1 table)

This paper contains 33 sections, 324 equations, 23 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kinetic Schemes for Hyperbolic Conservation Equations
  3. Strategy for flexible velocity kinetic schemes
  4. Flexible Velocity Boltzmann Scheme for 1-D convection equation
  5. 1-D convection equation in moment form
  6. Dirac delta based equilibrium distribution and flexible velocities
  7. Flexible Velocity Boltzmann Equation and Moments
  8. Kinetic Flux Splitting Based on Flexible Velocities
  9. Fixing Lambda
  10. Chapman-Enskog type expansion of the flexible velocity Boltzmann equation
  11. Fixing Lambda based on R-H (jump) condition
  12. Stability Analysis
  13. Second order accuracy
  14. Kinetic Lax-Wendroff (KLW) Method
  15. High resolution TVD KFDS Method
  16. ...and 18 more sections

Figures (23)

  • Figure 1: 3-point stencil for a FVM
  • Figure 2: Shock at a cell-interface in FVM
  • Figure 3: L1 Norm Errors for a) KFDS b) KFDS+ c) KLW d) TVD-KFDS & e) TVD-KFDS+ schemes
  • Figure 4: L2 Norm Errors for a) KFDS b) KFDS+ c) KLW d) TVD-KFDS & e) TVD-KFDS+ schemes
  • Figure 5: Test Case 1 : KFDS, KLW & TVD-KFDS schemes for linear convection
  • ...and 18 more figures