On the Stability of Multi-Step Finite-Difference-Based Lattice Boltzmann Schemes

2018 ◽  
Vol 16 (01) ◽  
pp. 1850087 ◽  
Author(s):  
Gerasim V. Krivovichev ◽  
Sergey A. Mikheev
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Time Derivative ◽  
Kinetic Equations ◽  
High Order ◽  
Numerical Characteristic ◽  
Central Difference ◽  
Boltzmann Equations ◽  
Von Neumann ◽  
The Stability

Stability of finite-difference-based off-lattice Boltzmann schemes is analyzed. The time derivative in system of discrete Boltzmann equations is approximated by two-step modified central difference. Advective term is approximated by finite differences from first- to fourth-orders of accuracy. Characteristics-based (CB) schemes and schemes with traditional separate approximations of space derivatives are considered. A special class of high-order CB schemes with approximation in the internal nodes of grid patterns is constructed. It is demonstrated that apparent viscosity for the schemes of high-order is equal to kinematic viscosity of the system of Bhatnaghar–Gross–Krook kinetic equations. Stability of the schemes is analyzed by the von Neumann method for the cases of two flow regimes in unbounded domain. Stability is analyzed by the investigation of the stability domains in parameter space. The area of the domain is considered as the main numerical characteristic of the stability. As the main result of the analysis, it must be mentioned that the areas of CB schemes are greater than areas for the schemes with separate approximations.

scholarly journals Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes

2019 ◽  
pp. 116-127
Author(s):  
Г.В. Кривовичев ◽  
М.П. Мащинская
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Lattice Boltzmann ◽  
Kinetic Equations ◽  
Finite Difference Schemes ◽  
Courant Number ◽  
Von Neumann ◽  
Upwind Schemes ◽  
Implicit Finite Difference ◽  
The Stability

Статья посвящена анализу устойчивости неявных конечно-разностных схем для системы кинетических уравнений, применяемых для проведения гидродинамических расчетов в рамках метода решеточных уравнений Больцмана. Представлены семейства двухслойных и трехслойных схем с направленными разностями первого-четвертого порядков аппроксимации по пространственным переменным. Важной особенностью схем является то, что конвективные слагаемые аппроксимируются одной конечной разностью. Показано, что в выражении для аппроксимационной вязкости схем высоких порядков отсутствуют фиктивные слагаемые, что позволяет применять их во всем диапазоне значений времени релаксации. Анализ устойчивости проводится по линейному приближению с использованием метода Неймана. Получены приближенные условия устойчивости в виде неравенств на значения параметра Куранта. При расчетах показано, что площади областей устойчивости в пространстве параметров у двухслойных схем больше, чем у трехслойных. Исследованные схемы могут применяться при расчетах как непосредственно, так и в методах типа предиктор-корректор. 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.

On the stability of lattice boltzmann equations for one-dimensional diffusion equation

2017 ◽  
Vol 08 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Gerasim Vladimirovich Krivovichev
Keyword(s):  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Kinetic Equations ◽  
Differential Approximation ◽  
Boltzmann Equations ◽  
Von Neumann ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Cauchy Problem ◽  
Lattice Boltzmann Equations

Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar–Gross–Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.

scholarly journals Analysis and optimization of higher order explicit finite-difference schemes for the advection stage implementation in the lattice Boltzmann method

2017 ◽  
pp. 227-246
Author(s):  
Г.В. Кривовичев ◽  
Е.С. Марнопольская
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Kinetic Equations ◽  
Difference Schemes ◽  
Numerical Dispersion ◽  
Finite Difference Schemes ◽  
Maximum Function ◽  
Driven Cavity ◽  
Explicit Finite Difference ◽  
The Stability

Статья посвящена анализу и оптимизации явных разностных схем для решения уравнений переноса, возникающих на этапе адвекции метода расщепления по физическим процессам. Метод может применяться как для решеточных уравнений Больцмана, так и при решении кинетических уравнений общего вида. Рассматриваются схемы второго-четвертого порядков аппроксимации. Для уменьшения эффектов численных диссипации и дисперсии используются схемы с параметром. С использованием метода фон Неймана и полиномиальной аппроксимации границ областей устойчивости получены условия устойчивости схем в виде неравенств на значения параметра Куранта. Оптимальные значения параметра для регулирования диссипативных и дисперсионных эффектов предлагается находить посредством решения задач минимизации функций максимума. Схемы с оптимальными значениями параметра применяются при решении тестовых задач - для одномерного и двумерного уравнений переноса, а также при применении метода расщепления к решению задачи о течении в каверне с подвижной крышкой. This paper is devoted to the analysis and optimization of explicit finite-difference schemes for solving the transport equations arising at the advection stage in the method of splitting into physical processes. The method can be applied to the lattice Boltzmann equations and to the kinetic equations of general type. The second-to-fourth order schemes are considered. In order to minimize the effect of numerical dispersion and dissipation, the parametric schemes are used. The Neumann method and the polynomial approximation of the boundaries of stability domains are employed to obtain the stability conditions in the form of inequalities imposed on the Courant parameter. The optimal values of the parameter used to control the dissipation and dispersion effects are found by minimizing the maximum function. The schemes with optimal parameters are applied for the numerical solution of 1D and 2D advection equations and for the problem of lid-driven cavity flow.

scholarly journals Stability study of finite-difference-based lattice Boltzmann schemes with upwind differences of high order approximation

2015 ◽  
pp. 196-204
Author(s):  
Г.В. Кривовичев ◽  
С.А. Михеев
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Linear Approximation ◽  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Order Approximation ◽  
Stability Study ◽  
High Order Approximation ◽  
Spatial Variables ◽  
The Stability

Исследуется устойчивость трехслойных конечно-разностных решеточных схем Больцмана третьего и четвертого порядков аппроксимации по пространственным переменным. Проводится анализ устойчивости по начальным условиям с использованием линейного приближения. Для исследования используется метод Неймана. Показано, что устойчивость схем можно улучшить за счет аппроксимации конвективных членов во внутренних узлах сеточного шаблона. В этом случае удается получать большие по площади области устойчивости, чем при аппроксимации в граничных узлах шаблона. The stability of three-level finite-difference-based lattice Boltzmann schemes of third and fourth orders of approximation with respect to spatial variables is studied. The stability analysis with respect to initial conditions is performed on the basis of a linear approximation. These studies are based on the Neumann method. It is shown that the stability of the schemes can be improved by the approximation convective terms in internal nodes of the grid stencils in use. In this case the stability domains are larger compared to the case of approximation in boundary nodes.

An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Joaquín Quintana Murillo ◽  
Santos Bravo Yuste
Keyword(s):  
Fractional Derivative ◽  
Difference Method ◽  
Wave Equations ◽  
Time Derivative ◽  
Fourier Method ◽  
Its Region ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Von Neumann ◽  
The Stability

An explicit difference method is considered for solving fractional diffusion and fractional diffusion-wave equations where the time derivative is a fractional derivative in the Caputo form. For the fractional diffusion equation, the L1 discretization formula of the fractional derivative is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. In both equations, the spatial derivative is approximated by means of the three-point centered formula. The accuracy of the present method is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a kind of fractional von Neumann (or Fourier) method. The stability bound so obtained, which is given in terms of the Riemann zeta function, is checked numerically.

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