scholarly journals Application of predictor-corrector finite-difference-based schemes in the lattice Boltzmann method

2015 ◽  
pp. 10-17
Author(s):  
Г.В. Кривовичев ◽  
Е.В. Воскобойникова
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Kinetic Equations ◽  
Taylor Vortex ◽  
Finite Difference Schemes ◽  
Time Step ◽  
Computation Process ◽  
Wide Range ◽  
Explicit Finite Difference ◽  
Predictor Corrector

Построены конечно-разностные решеточные схемы Больцмана типа предиктор-корректор. Рассмотрены подход с раздельной аппроксимацией пространственных производных в конвективных членах кинетических уравнений и подход, когда эти члены заменяются одной конечной разностью. На обоих этапах процесса вычислений на одном шаге используются явные разностные схемы. При решении задачи о течении в каверне и задачи о вихрях Тейлора в широком диапазоне изменения числа Рейнольдса показано, что построенные схемы позволяют проводить расчеты с большим значением шага по времени, чем некоторые другие известные схемы. Predictor-corrector finite-difference-based lattice Boltzmann schemes are proposed. An approach with separate approximation of spatial derivatives in the convective terms of kinetic equations and an approach when these terms are replaced by a single finite difference are considered. Explicit finite-difference schemes are used at both the stages of the computation process. The cavity flow problem and the Taylor vortex problem are solved numerically in a wide range of the Reynolds number. It is shown that the proposed schemes allow a larger time step compared to other known schemes.

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.

IMPLICIT-EXPLICIT FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD FOR COMPRESSIBLE FLOWS

2007 ◽  
Vol 18 (12) ◽  
pp. 1961-1983 ◽  
Author(s):  
Y. WANG ◽  
Y. L. HE ◽  
T. S. ZHAO ◽  
G. H. TANG ◽  
W. Q. TAO
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Compressible Flows ◽  
Time Discretization ◽  
Taylor Vortex ◽  
Lattice Boltzmann Methods ◽  
Taylor Vortex Flow ◽  
Explicit Finite Difference ◽  
Boltzmann Method

We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge–Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant–Friedriche–Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.

Removing the stability limit of the time–space domain explicit finite difference schemes for acoustic modeling with stability condition-based spatial operators

Geophysics ◽  
2021 ◽  
pp. 1-82
Author(s):  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Grid Spacing ◽  
Time Step ◽  
Space Domain ◽  
Wavenumber Range ◽  
Time Space ◽  
Explicit Finite Difference ◽  
The Stability

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time–space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. Additionally, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis and numerical modeling demonstrate the accuracy and effectiveness of the proposed method.

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.

Paper 3: Recent Developments in Fluid Film Lubrication Theory

1967 ◽  
Vol 182 (1) ◽  
pp. 36-50 ◽  
Author(s):  
T. Lloyd ◽  
H. McCallion
Keyword(s):  
Finite Difference ◽  
High Speed ◽  
Lubrication Theory ◽  
Fluid Film ◽  
Special Topic ◽  
Finite Difference Schemes ◽  
Great Activity ◽  
Wide Range ◽  
Film Lubrication ◽  
Fluid Film Lubrication

Developments in high-speed electronic computers have greatly influenced the progress in fluid film lubrication over the past ten years. Static and dynamic oil film parameters have been computed for a wide range of finite geometries, for hydrostatic and hydrodynamic bearings lubricated by compressible and incompressible lubricants. These are either sufficient in themselves or else act as a yardstick against which approximate formulas may be tested. Much use has been made of iterative finite difference schemes, which are particularly well suited to digital computers, and these methods are now more fully understood. Other methods of solution include direct inversion of finite difference matrices and solution by expression of the pressure by some infinite series, a finite number of terms of which give adequate representation. Besides the increase in design data available, there has been substantial progress through a re-examination of the effects of modifying some of the assumptions inherent in most of the available solutions of the Reynolds equation. These include the assumption of constant lubricant viscosity, of rigid surfaces and of laminar flow. Major progress has been witnessed in two fields. The interaction of the lubricant film with elastic boundaries has been shown to be of prime importance in highly loaded contacts such as gears. This has led to the development of the special topic of elastohydrodynamic lubrication theory. The applicability of gas bearings in such growing industries as computers, space vehicles and nuclear reactors has resulted in great activity and progress in this field.

Predictor-corrector finite-difference lattice Boltzmann schemes

2015 ◽  
Vol 9 ◽  
pp. 4191-4199 ◽  
Author(s):  
G. V. Krivovichev ◽  
E. V. Voskoboinikova
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Predictor Corrector

Prefactored optimized compact finite-difference schemes for second spatial derivatives

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB87-WB95 ◽  
Author(s):  
Hongbo Zhou ◽  
Guanquan Zhang
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Compact Scheme ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Compact Schemes ◽  
Compact Finite Difference ◽  
Wide Range ◽  
Compact Finite Difference Schemes ◽  
Spatial Derivatives

We have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives. First, instead of emphasizing high resolution of a single monochromatic wave, we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers. This leads to the development of the optimized compact schemes whose coefficients will be determined by Fourier analysis and the least-squares optimization in the wavenumber domain. The resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations. However, solving for each spatial derivative using these compact schemes requires the inversion of a band matrix. To resolve this issue, we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes, which can be solved explicitly. We achieve this by ensuring a property that the real numerical wavenumbers of both the forward and backward biased stencils are the same as that of the original central compact scheme, and their imaginary numerical wavenumbers have the same values but with opposite signs. This property guarantees that the original optimized compact scheme can be completely recovered after the summation of the forward and backward finite difference operators. These prefactored optimized compact schemes have smaller stencil sizes than even those of the original compact schemes, and hence, they can take full advantage of the computer caches without sacrificing their resolving power. Comparisons were made throughout with other well-known schemes.

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