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Abstracts

INTRODUCTION

The lattice Boltzmann method (LBM) is a relatively new and promising numerical technique for simulating a broad variety of complex physical systems, especially fluid flows [1-4]. Having arisen about twenty years ago, the LBM is becoming a serious alternative to traditional numerical methods, such as finite-difference/volume/ element schemes.

In this report the lattice Boltzmann method is applied for two-dimensional shallow water equations (SWE) describing tidal flows in seas and oceans. The exact kind of the SWE is presented in [5] where this equations have been solved by the finite element method. The technique of using the LBM for the SWE is presented for the first time in [3]. Here, unlike [3], we consider another type of shallow water equations and also a parallel implementation is shown. Some tests of the parallel code, performed on the cluster MVS-100K (about 8000 CPU, 95 Tflops), have given satisfactory results.

LATTICE BOLTZMANN METHOD

The basic idea of the LBM is to use a simplified kinetic Boltzmann equation of a gas which is nevertheless capable on the macroscopic level to give correct average values for velocity, density, pressure and for other characteristics of fluid flows. Unlike the traditional methods, based on discretizations of continuum conservation equations (mass, momentum, energy), the LBM models the fluid consisting of fictive mesoscopic particles, and such particles perform consecutive "free flight" and collision processes over a discrete lattice grid. So, the lattice Boltzmann model is based on the statistical physics and describes the microscopic behavior of particles in a very simplified manner, but it correctly describe the macroscopic flow behavior.

The main advantages of the LBM are that: (1) initial equations have a simple form; there are derivatives only of the first order; the convection operator is linear; nonlinearity is present only in an algebraic source term; (2) in view of local character of calculations (only nearby particles interact with each other), the LB-method is easily realised on parallel computers; (3) the pressure is calculated using the equation of state; there is no necessity to solve the Poisson equation in all domain; (4) it is a convenient and perspective tool for modeling of physical and chemical processes in geometrically complex areas of micro and nano sizes (porous media and nano-structures); (5) easy of incorporating microscopic interactions and boundary conditions; (6) simplicity of programming.

PARALLELIZATION USING DVM-SYSTEM

A parallel version of the LBM for the shallow water equations has been implemented in this work using the Fortran-DVM language developed in the Keldysh Institute of Applied Mathematics of RAS [6-7]. The main goals of Fortran-DVM are follows [7]. Simplicity of parallel program development. Portability of parallel program onto different architecture computers (serial and parallel). For serial computers the portability is provided by DVM-directive "transparency" for standard Fortran 77 compilers. High performance of program execution. Unified parallelism model for Fortran 77 languages, and, as result, unified system of runtime support, debugging, performance analyzing and prediction. Domain decomposition has been performed to parallelize the lattice Boltzmann method.

In the recent years the Fortran-DVM/OpenMP language also has been developed in the Keldysh Institute. It allows essentially to automate a programming of SMP-clusters which are using multiprocessors.

NUMERICAL EXPERIMENTS

To verify the code, we solve some benchmark tasks. The numerical results are compared with either analytical solutions or numerical results reported in the literature. The first test is so called "tidal wave flow" [3]. This is 1-D problem with bed profile prescribed exactly. The analytical solution of this task is given in [8]. Other test concerns distributions of surface waves in 2-D square area. At initial moment the "bulb" is set at the centre of area. Solutions of this problem by the LBM and FEM [5] are compared.

This work is supported by the Russian Foundation for Basic Research (Grant No. 08-01-00621).

REFERENCES

[1]. S.Chen, G.D.Doolen. Lattice Boltzmann methods for fluid flows. Annu. Rev. Fluid Mech. vol 30, 329-364, 1998. [2]. S.Succi. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, 2001. [3]. J.G.Zhou. A lattice Boltzmann model for the shallow water equations. Comput. Methods Appl. Mech. Eng. vol 191, 3527-3539, 2002. [4]. M.C.Sukop, D.T.Thorne. Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. Springer, 2007. [5]. L.P.Kamenshchikov, E.D.Karepova, V.V.Shaidurov. Simulation of surface waves in basin by the finite elements method. Russ. J. Numer. Anal. Math. Modelling, vol. 21, 305-320, 2006. [6]. V.A.Krukov. Working out of Parallel Programs for Computing Clusters and Networks. The Information Technology and Computing Systems (in Russian). No. 1-2, 42-61, 2003. [7]. DVM-System, http://www.keldysh.ru/dvm. [8]. A.Bermudez, M.E.Vazquez. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids, vol. 23, 1049-1071. 1994

Note. Abstracts are published in author's edition

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