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.

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.

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.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

2016 ◽  
Vol 27 (09) ◽  
pp. 1650097 ◽  
Author(s):  
A. H. Encinas ◽  
V. Gayoso-Martínez ◽  
A. Martín del Rey ◽  
J. Martín-Vaquero ◽  
A. Queiruga-Dios
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Nonlinear Phenomena ◽  
Klein Gordon ◽  
Implicit Finite Difference ◽  
The Stability ◽  
Stability And Consistency ◽  
New Algorithms

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

scholarly journals Stability theory for some scalar finite difference schemes: validity of the modified equations approach

2021 ◽  
Vol 70 ◽  
pp. 124-136
Author(s):  
Firas Dhaouadi ◽  
Emilie Duval ◽  
Sergey Tkachenko ◽  
Jean-Paul Vila
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Finite Difference ◽  
Heat Equation ◽  
Infinite Series ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Modified Equations ◽  
The Stability ◽  
Modified Equation

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.

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.

scholarly journals A Stability Analysis of Finite-Volume Advection Schemes Permitting Long Time Steps

2007 ◽  
Vol 135 (7) ◽  
pp. 2658-2673 ◽  
Author(s):  
Peter Hjort Lauritzen
Keyword(s):  
Stability Analysis ◽  
Finite Volume ◽  
Two Dimensional ◽  
Finite Volume Schemes ◽  
Courant Number ◽  
Von Neumann ◽  
Dispersion Properties ◽  
Advection Schemes ◽  
Long Time ◽  
The Stability

Abstract Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

scholarly journals Stability analysis of finite difference schemes for an advection diffusion equation

2017 ◽  
Vol 29 (2) ◽  
pp. 143-151 ◽  
Author(s):  
TMAK Azad ◽  
LS Andallah
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Time Step ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
The Stability ◽  
Stability Results

The paper studies stability analysis for two standard finite difference schemes FTBSCS (forward time backward space and centered space) and FTCS (forward time and centered space). One-dimensional advection diffusion equation is solved by using the schemes with appropriate initial and boundary conditions. Numerical experiments are performed to verify the stability results obtained in this study. It is found that FTCS scheme gives better point-wise solutions than FTBSCS in terms of time step selection.Bangladesh J. Sci. Res. 29(2): 143-151, December-2016

A stability checking procedure for finite-difference schemes with boundary conditions in acoustic media

1989 ◽  
Vol 79 (5) ◽  
pp. 1601-1606
Author(s):  
Aladin H. Kamel
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Stability Condition ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Neumann Condition ◽  
Time Behavior ◽  
Von Neumann ◽  
The Stability ◽  
Stability And Accuracy

Abstract The manner in which boundary conditions are approximated and introduced into finite-difference schemes has an important influence on the stability and accuracy of the results. The standard von Neumann stability condition applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. In this paper, we extend the von Neumann condition to include boundary conditions. The method is based on studying the time propagating matrix which governs the space-time behavior of the numerical grid. Examples of applying the procedure on schemes with different boundary conditions are given.

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