On the Optimization of Numerical Dispersion and Dissipation of Finite Difference Scheme for Linear Advection Equation

Transcription

1 Applied Mathematical Sciences, Vol. 0, 206, no. 48, HIKARI Ltd, On the Optimization of Nmerical Dispersion and Dissipation of Finite Difference Scheme for Linear Advection Eqation G. V. Krivovichev Faclty of Applied Mathematics and Processes of Control Saint Petersbrg State University, 7/9 Universitetskaya nab., Saint Petersbrg, 99034, Rssian Federation E. S. Marnopolskaya Faclty of Applied Mathematics and Processes of Control Saint Petersbrg State University, 7/9 Universitetskaya nab., Saint Petersbrg, 99034, Rssian Federation Copyright c 206 G. V. Krivovichev and E. S. Marnopolskaya. This article is distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. Abstract Finite difference scheme for linear advection eqation with dependence on scalar dimensionless parameter is constrcted. Stability of the scheme is investigated by von Nemann method and stability domain in parameter space is constrcted. Dispersive and dissipative properties of the scheme are optimized by the choice of scalar parameter. Low dispersion and dissipation of the scheme is demonstrated by the nmerical soltion of simple test Cachy problem. Scheme constrcted may be applied in comptations based on splitting method for Boltzmann or lattice Boltzmann eqations. Mathematics Sbect Classification: 65M2, 35E99 Keywords: lattice Boltzmann method, advection, stability, dispersion, dissipation

2 2382 G. V. Krivovichev and E. S. Marnopolskaya Introdction Nowadays lattice Boltzmann method LBM) [2], [9], [], [2] is considered as a powerfl method for the soltion of different flid dynamics problems [3], [3], [4]. The method is based on the soltion of the system of kinetic eqations instead the system of the eqations of hydrodynamics. Some complex problems for mltiphase flows and flows in poros media were solved by this method [], [6]. One of the well-known comptational algorithm of LBM is based on the splitting on physical processes [5], [8]. Splitting process is realized on every time step by two stages collision and advection. On the advection step the soltion of the system for the particle advection process is realized. The process is described by system of linear hyperbolic eqations. On the collision stage nonlinear system withot space derivatives is solved [5]. In this stdy finite difference scheme FDS) for the soltion of linear advection eqation is considered. The scheme is dependent on dimensionless scalar parameter. The properties of the scheme sch as stability and existence of fictitios effects dispersion and dissipation) are investigated. The problems of the optimization of dispersive and dissipative srfaces are considered. Optimal parameter vale is obtained. The paper is organized as follows. In Section 2 the linear advection system and FDS are considered. In Section 3 the properties of FDS are investigated. Conclding remarks are made in Section 4. 2 FDS for linear advection system The system of linear advection eqations is written as: f i t + V i f i = 0, ) where f i = f i t, r), i =, n are the distribtion fnctions of the particles with velocities V i = V e i, V = l/δt, where l is a mean free path, δt is a mean free time, t is a time, r is a vector of space variables. System ) is solved on the advection stage of the method of splitting on physical processes in LBM or in soltion procedre for general Boltzmann eqation [4]. For the simplification of the analysis D variant of ) is considered: f i t + V f i i = 0, 2) x where i =, 2, e =, e 2 = DQ2 lattice). Dimensionless variables are introdced: t := t δt, x := x l, f i := f i, 3) Φ i

3 On the optimization of nmerical dispersion and dissipation of where Φ i are the mean vales of f i. After the sbstittion of 3) into 2) the following system is obtained: f i t + e f i i x = 0. 4) Eqations of 4) are independent on each other, so the case of linear scalar eqation may be considered: t + c x = 0, 5) where = f i, c = e i. For the discretization of 5) the following formlae for time derivative is considered: t t, x n ) t +, x n ) t 2, x n ) + t 2, x n )), 2 t where n t, x n ), where t is a time step, t is a time node, x n is a node of space grid constrcted with step h. Let s consider two approximations of the term c / x by central and first order pwind differences along the characteristics of eq. 5): c x t, x n ) c 2h t, x n + signc)h) t, x n signc)h)), c x t, x n ) c h t, x n ) t, x n signc)h)). For the simplification of the analysis the case of c > 0 is considered. So the following two-step FDS s for 5) are constrcted: + n = 2 + n = 2 ) n + 2 n γ n+ n ), 6) ) n + 2 n 2γ n ) n, 7) where γ = c t/h is a Corant nmber. The stability of the schemes 6) and 7) may be analyzed by von Nemann method. The soltions of 6) and 7) are presented in following form: n = λ ϕ)e inϕ, 8) where ϕ [0, 2π), i 2 =, λϕ) is a spectral fnction. After the sbstittion of 8) into 6) and 7) the following cbic eqations on λ are obtained: λ 3 + i2γ sinϕ) ) λ = 0,

4 2384 G. V. Krivovichev and E. S. Marnopolskaya λ 3 + i2γ e iϕ ) ) λ = 0. The roots of these eqations may be obtained by Cardano s formlas or in packages of compter algebra. The following stability conditions are obtained after analysis of absolte vales of the roots: γ /2 for 6) and γ /4 for 7). The reslts of the soltion of simple test Cachy problem for the case of c =, x [0, 0] and initial condition: 0, x) =, if x [4, 6] and 0, x) = 0 elsewhere, are presented at fig.. As it can be seen, the fictitios nmerical oscillations exists in the nmerical soltion obtained by scheme 6), this effect is known as a presence of nmerical dispersion. Nmerical soltion obtained by 7) tend to nll when t +, this effect is known as nmerical dissipation x Fig.. Soltions of the Cachy problem for eq. 5): nmerical soltion by FDS 6); 2 analytical soltion; 3 nmerical soltion by FDS 7). To avoid these fictitios effects, the scalar parameter ε [0, ] may be inclded to FDS. De to the idea of Go and Zhao [6], this parameter may be inclded by the following way: c x t, x n ) ε c ) UW x t, x n ) + ε) c ) C x t, x n ), where c t, x n )/ x) UW is approximation by first order pwind difference, c t, x n )/ x) C is approximation by central difference. The scheme obtained is presented by following formlae: + n = 2 ) n + 2 n 2γε n ) n γ ε) n+ n ). 9)

5 On the optimization of nmerical dispersion and dissipation of Investigation of properties of the schemes 3. Stability After the sbstittion of 8) into 9) the following cbic eqation on λϕ) is obtained: λ 3 + 2γε cosϕ)) + i sinϕ)) ) λ 2 = 0. 0) 2 2 As a reslt of nmerical analysis of roots of 0) stability domain of the scheme 9) in parameter space ε, γ) is constrcted. At fig. 2 the plot of bondary os this domain is presented Unstability Stability Fig. 2. Bondary of the stability domain of scheme 9). 3.2 Nmerical dispersion and dissipation Investigation of nmerical dispersion and dissipation and optimization of FDS 9) is based on the analysis of dispersive and dissipative srfaces. The idea of this approach is presented in [7]. Soltion of 9) is presented in form of traveling wave: n = e iω t knh), ) where ω is a freqency, k is a wave nmber. After the sbstittion of ) into 9), the following eqation on q = e iω t is obtained: q 3 + 2γε cosξ)) i sinξ)) ) q 2 = 0, 2) 2 2

6 2386 G. V. Krivovichev and E. S. Marnopolskaya where ξ = kh. There are three roots of eq. 2): q s, s =, 3. The real parts of freqencies ω s, which are the main characteristics of nmerical dispersion, may be expressed from q s by following formlas: ψ s γ, ξ, ε) = Reω s ) = arctan ) Imq s) Req s). 3) t Dispersive relation for eq. 5) is presented as: ω = ck. 4) Expressions for dispersive srface of eq. 5) may be obtained from 4) as a fnction of γ and ξ: ωγ, ξ) = γξ t. 5) De to the dependence of ψ s and ω see eqs. 3) and 5)) on γ, the case of t = may be considered. Optimal dispersive properties of the scheme 9) may be characterized by following fnction: ) Iε) = sp s=,2,3 sp ψ s γ, ξ, ε) ωγ, ξ) γ,ξ) Optimal vale of parameter ε is considered as a soltion of minimization problem of Iε). This fnction is minimized on the domain {γ, ξ) γ 0, γ ε), ξ [ π, π]}, where γ ε) is an pper bondary of stability domain at fixed ε. For the characterization of dissipative properties eq. ) is rewritten as: n = q e iargq ) knh). 6) As it can be seen from 6), dissipative property is characterized by fnctions η s γ, ξ, ε) = q s. For the obtaining of optimal vale of ε, the fnction F ε), presented as: ) F ε) = sp s=,2,3 sp η s γ, ξ, ε) C s γ,ξ)., 7) is minimized. In eq. 7) C s are constants, which are the vales of η s in case of low dissipation this case characterized by small vales of γ). Plots of Iε) and F ε) are presented at fig. 3 and fig. 4 respectively. As it can be seen, optimal parameter for Iε) is eqal to nity, while minimm of F ε) is realized in internal point of interval [0, ] in ε It is easy to concern see fig. ) that the main inflence on nmerical soltion is realized by nmerical dissipation, so the optimal parameter vale is chosen to be eqal to 0.5.

7 On the optimization of nmerical dispersion and dissipation of Fig. 3. Plot of Iε) Fig. 4. Plot of F ε) x Fig. 5. Plot of nmerical soltion of the test Cachy problem obtained by scheme 9) at optimal parameter vale: analytical soltion, 2 nmerical soltion.

8 2388 G. V. Krivovichev and E. S. Marnopolskaya At fig. 5 the plot of nmerical soltion, obtained by scheme 9) with optimal parameter is presented. As it can be seen, nmerical soltion demonstrate the absence of fictitios oscillations. At the same time, the amplitde of the soltion is close to constant and not decreased intensively. 4 Conclsion FDS for linear advection eqation with dependence on scalar dimensionless parameter is constrcted. Stability of the scheme is investigated by von Nemann method and stability domain in parameter space is constrcted. Dispersive and dissipative properties of the scheme are optimized by the choice of scalar parameter. Scheme constrcted may be applied in comptations based on splitting method for Boltzmann or lattice Boltzmann eqations. Acknowledgements. The reported stdy was fnded by Rssian Fondation of Basic Research according to research proect No mol a. References [] D.A. Biklov, A.A. Saratov and E.A. Grachev, Prediction of the permeability of proppant packs nder load, International Jornal of Modern Physics C, ), [2] S. Chen and G. D. Doolen, Lattice Boltzmann method for flid flows, Annal Review of Flid Mechanics, ), [3] N. N. Ermolaeva and G. I. Krbatova, The model of ice growth on the oter srface of a cylinder in sea water, 205 International Conference on Stability and Control Processes in Memory of V.I. Zbov, 205), [4] F. Filbet and G. Rsso, High order nmerical methods for the space nonhomogeneos Boltzmann eqation, Jornal of Comptational Physics, ), [5] Z. Go, C. Zheng and T. S. Zhao, A lattice BGK scheme for general propagation, Jornal of Scientific Compting, 6 200), no. 4,

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