Geodesic
Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes
Numerical methods and programming, Tome 20 (2019) no. 2, pp. 116-127.

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The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.
Keywords: lattice Boltzmann method, implicit finite-difference schemes, stability. 
@article{VMP_2019_20_2_a3,
     author = {G. V. Krivovichev and M. P. Mashchinskaya},
     title = {Stability analysis of the implicit finite-difference-based upwind lattice {Boltzmann} schemes},
     journal = {Numerical methods and programming},
     pages = {116--127},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2019_20_2_a3/}
}
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G. V. Krivovichev; M. P. Mashchinskaya. Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes. Numerical methods and programming, Tome 20 (2019) no. 2, pp. 116-127. http://geodesic.mathdoc.fr/item/VMP_2019_20_2_a3/