Numerical Solution of the Steady Convection-diffusion ... - ScienceDirect

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Procedia Computer Science 18 (2013) 2095 – 2100

International Conference on Computational Science, ICCS 2013

Numerical Solution of the Steady Convection-Diffusion Equation with Dominant Convection L.A. Krukier∗, O.A. Pichugina, B.L. Krukier Southern Federal University, Computer Center, 200/1 Stachki Ave., Bld.2, Rostov-on-Don, 344090, Russia

Abstract Steady convection-diffusion equation in 2-D domain is considered. Central finite-difference approximation has been taken to obtain a large sparse nonsymmetric linear system with positive real matrix. New class of product triangular skew-symmetric iterative methods for solution of such system is presented and considered. Using this method as preconditioner for GMRES and BiCG has been made. Results of numerical experiments for two-dimensional convection-diffusion equation for different big Peclet numbers and velocity coefficients have been presented.

© by Elsevier OpenPublished access under BY-NC-ND c 2013  2013The L.A.Authors. Krukier,Published O.A. Pichugina, B.L. B.V. Krukier. by CC Elsevier B.V. license. Selection peerpeer-review review under responsibility of the of theof2013 International Conference on Computational Selection and and/or under responsibility of organizers the organizers the 2013 International Conference on Computational Science Science. Keywords: convection-diffusion problem; product triangular iterative method; preconditioners

1. Introduction Mathematical modeling of convection-diffusion problems is the most widespread in the science and engineering. Very often the dimensionless parameter that measures the relative strength of the diffusion is quite small, so one often meets with situations where boundary and interior layers are present and singular perturbation problems arise [14]. The small parameter at the higher derivatives is necessary but not a sufficient condition to appear of boundary layer [12]. It is necessary to add property that boundary conditions and right-hand side of equation don’t conform to each other. In such circumstances difficulties will be experienced with standard numerical approximation. It is also due to the fact that numerical algorithms and the techniques used for their analysis tend to be very different in the two limiting cases of elliptic and hyperbolic equations. The choice of the discretization method is very important question for the partial differential equation with first order derivatives and the most interesting moment of the solution the problem. It is well known [17, 14] that in general linear equation systems with M-matrix [16] can be obtained by application of upwind schemes while positive real matrix can be obtained by using central FD schemes [10]. Each one of these schemes has their own advantages and deficiencies which have been discussed in [18, 14]. ∗ Corresponding

author. Tel.: +7-863-219-9710; fax: +7-863-219-9711. E-mail address: [email protected].

1877-0509 © 2013 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selection and peer review under responsibility of the organizers of the 2013 International Conference on Computational Science doi:10.1016/j.procs.2013.05.379

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It is well known [7], when convection-diffusion problems with boundary or interior layers are discretized using the central difference method or the finite element method in classical formulation, the numerical solution is polluted with oscillations because the corresponding difference operator is, in general, not monotone. An alternative approach is to approximate the first derivatives using some upwind technique or artificial diffusion [14, 18, 8]. One then obtains a monotone operator and oscillations free numerically computed solution but with lower accuracy of the approximation scheme and smearing of the boundary layers. Convection-diffusion equation is a good test for iterative methods. A lot of papers [3, 6, 11, 13, 18] described numerical experiments with convection-diffusion equation for different parameters. Consider the convection-diffusion equation in bounded domain Ω writing in different forms:

or

LNC u = −Pe−1 L2 u + L1ND u + L0 u − F, ∂u ∂u L1 u = K1 + K2 , Ki = Ki (x, y), i = 1, 2 ∂x ∂y

(1)

LC u = −Pe−1 L2 u + L1D u + L0 u − F, ∂ ∂ L1 u = (K1 u) + (K2 u), Ki = Ki (x, y), i = 1, 2 ∂x ∂y

(2)

where u |∂Ω = u |b , is boundary condition and L2 u =

2  α,β=1

Lαβ u,

Lαβ u =

∂u ∂ (Kαβ ), ∂xα ∂xβ

Kαβ = Kαβ (x, y),

α, β = 1, 2 (3)

L0 u = K0 u, K0 = K0 (x, y) ≥ 0, u = u(x, y), F = F(x, y), (x, y) ∈ Ω = [0, 1] × [0, 1]

are common coefficients for (1), (2), Pe is the Peclet number, K = {K1 , K2 } is the velocity vector, F is right-hand side, u is the solution and u |b is a function defined on the domain boundary. Convective terms in (1) are written in nonconservative form, whereas (2) contain them in conservative form. One can rewrite (1) and (2) as follows: Lu = −Pe−1 L2 u + L1S u + LN0 u − F,  ∂u ∂u 1 ∂ ∂ K1 L1S u = + K2 + (K1 u) + (K2 u) , 2 ∂x ∂y ∂x ∂y Ki = Ki (x, y), i = 1, 2, 1 ∂K1 ∂K2 LN0 = L0 + σ(divK), divK = + . 2 ∂x ∂y

(4)

where σ = −1 for equation (1) and σ = 1 for equation (2). When the medium is incompressible, i.e. the condition divK = 0

(5)

holds, then forms (1) and (2) are equivalent and the convection-diffusion equation can be written in the so-called ”symmetric” form [10, 13]. LS u = −Pe−1 L2 u + L1S u + L 0 u − F,  ∂u ∂u 1 ∂ ∂ K1 L1S u = + K2 + (K1 u) + (K2 u) , Ki = Ki (x, y), 2 ∂x ∂y ∂x ∂y

i = 1, 2,

Additionally assume that condition of ellipticity for coefficients from (3) is fulfilled [14].

(6)

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2. Finite difference approximation of the equation and system of linear algebraic equation The uniform grid Ωh with step h x = hy = h in domain Ω has been introduced. The boundary conditions on ∂Ω are interpolated on the boundary ∂Ωh with a second order truncation error. Coefficients of difference scheme include the quantity S S C = Peh/2 (7) which was called the skew-symmetry coefficient of the problem. Using natural ordering of unknowns, transform difference scheme to the nonsymmetric nine-diagonals linear system of equations Au = f, 1 A = AΔ + A1 + D, A0 = (A + A∗ ) = AΔ + D = A∗0 , 2 1 A1 = (A − A∗ ) = −A∗1 , 2

(8)

where A is (N − 1) × (N − 1) matrix, N = 1/h, u = {u11 , u12 , ..., uN−1N−1 }T is the vector of solution, f = { f11 , f12 , ..., fN−1N−1 }T is the vector of the right part. Operator A can be naturally expressed [10] in the case of central difference approximation of the convective terms in (4) as a sum of symmetric positive definite operator AΔ , skew-symmetric operator A1 and diagonal operator D. AΔ is a difference analogue L2h of operator L2 , describing a diffusion process, D is discrete analogue of sources and drains terms in the equation (4), described by operator K0 and artificial term 1/2σ(divK). A1 is a difference analogue of the convective terms. Thus, linear system (8) with non-symmetric matrix A is constructed. If in (8) A0 is a positive definite matrix, then matrix A is called positive real. The linear system (8) is called strongly non-symmetric if A0  / A1  ∼ O(1), where ∗ is one of matrix norms. 3. Iterative solution Different basic iterative methods such as ILU [15], SOR [17] and ADI [3] have been used directly for solution of arising linear equation systems as well as preconditioners for CG or BiCG type’s methods [15]. As it was shown in [1], ILU as preconditioner for GMRES(20) and Bi-CGSTAB has been broken for large S S C from (7) and natural ordering of unknowns. ADI [3] and block SOR [4] have been quite successful for the case when Rh is near unity. We present new type of two parameters product triangular (PTM) iterative methods that use the skew-symmetric part of the matrix as an input and only require the matrix to be positive real. Using this class of iterative methods as preconditioners for GMRES(m) and BiCG are considered. Some ideas using symmetric – skew-symmetric splitting of the matrix to solve linear equation systems have been firstly proposed in [9], but idea to split of skew-symmetric part of an initial matrix has been proposed firstly in [10]. Let us approach (8) by considering the iterative method of the following form: yn+1 − yn (9) + Ayn = f, n ≥ 0 τ where f, y0 ∈ H, H is an n-dimensional real Hilbert space, f is the right part of (8), A, B(ω) are linear operators (matrices) in H, A is given by equation (8), B(ω) is invertible, y0 is an initial guess, yk is the k-th approach, τ, ω > 0 are iterative parameters, u is the solution that we obtain, ek = yk − u and rk = Aek denote the error and the residual in the k-th iteration, respectively. Method (9) may be also represented as yn+1 = Gyn + τ f, B(ω)

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G = B−1 (ω)(B(ω) − τA).

(10)

Consider choice of operator B for PTM B(ω) = (BC + ωKU )BC−1 (BC + ωKL ),

BC = BC∗

(11)

where KL + KU = A1 , KL = −KU∗ , BC = BC∗ . Operators KL and KU represent strictly upper and lower triangular part of a skew-symmetric matrix A1 from (8) and operator BC can be chosen arbitrarily, but has to be symmetric. Choice B as in (11) was for the first time introduced for ω = τ in [2]. Sufficient condition of convergence for PTM has been proved in [12]. As a matter of fact, many widespread preconditioners are very symmetric by nature, whereas frequently it does not need to improve ”features” of symmetric part and the main source of difficulties is concealed in large and/or poor-conditioned nonsymmetric part. Generally speaking, we show that the proposed technique is primary destined for strongly nonsymmetric systems, and probably, especially for ones whose symmetric part is well-conditioned and does not require preconditioning. 4. PTM as preconditioner for GMRES(10) and BiCG Equation (1) is discretized by centered differences for both the Laplacian and the first derivatives on a uniform grid with 31 × 31 interior nodes. Linear systems arising from the standard 5-point FD approximation of the steady convection-diffusion problem of the form (1) with K12 = K21 = 0

(12)

where F is chosen so that the solution of (1) is defined as u˜ (x, y) = e xy sin πx sin πy is considered. As one can easily verify, while discretization in (1), convective terms contribute to matrix A exactly its skewsymmetric part A1 . The used velocity coefficients of (1) are presented in the Table 1. Table 1. Velocity coefficients for test problems. Problem No. 1 2 3 4

v1 1 1 − 2x x+y sin 2πx

v2 −1 2y − 1 x−y −2πy cos 2πx

We set Pe = 103 , 104 and 105 in numerical tests. As one can see, for each model problem they are chosen to satisfy the constraint div(v) = v1 x + v2y = 0 (which follows from the medium incompressibility for the problem (1)). On the whole, in order to the test results be comparable with those obtained in the other adjacent papers we take the analytical solution and the velocity coefficients similar to those in [5]. The initial guess in all runs was a zero vector and iterations were performed until rm /r0  ≤ 10−6 ,

(13)

where rm is the residual vector, and   represents the Euclidean norm. For each test the number of iterations needed to satisfy (13) is presented in the tables. Variants of PTM named PTMBc(1) and PTMBc(3) determinate [11] special choice of matrix BC and iterative parameter τ.

L.A. Krukier et al. / Procedia Computer Science 18 (2013) 2095 – 2100 Table 2. GMRES(10) with PTM. Pe GMRES(10) Problem 1, v=1, u=-1 103 28 211 104 1473 105 Problem 2, v=1-2x, u=2y-1 103 33 179 104 105 609 Problem 3, v=x+y,u=x-y 103 39 210 104 1567 105 Problem 4,v=sin2πx, u=-2πycos2πx 103 77 557 104 4515 105

Table 3. BiCG with PTM. Pe BiCG Problem 1, v=1, u=-1 103 139 421 104 105 767 Problem 2, v=1-2x, u=2y-1 103 99 187 104 209 105 Problem 3, v=x+y, u=x-y 103 188 1543 104 2496 105 Problem 4, v=sin2πx, u=-2πycos2πx 103 434 1524 104 105 1606

2099

GMRES(10)+PTMBc(1)(τ)

GMRES(10)+PTM(3)(τ)

GMRES(10)+SSOR

6 42 161

8 36 277

11 42 314

6 33 224

4 21 123

15 44 321

9 47 295

6 17 119

23 53 315

19 133 859

6 19 132

28 200 2242

BiCG+PTM(1)(τ)

BiCG+PTMBc(3)(τ)

BiCG+SSOR

75 447 657

137 594 756

60 308 789

64 232 475

98 272 235

61 259 520

81 390 1065

211 1670 1484

69 365 1091

166 689 943

289 658 674

161 709 1151

5. Conclusions For iterative solution processes of linear systems with strongly nonsymmetric matrices we propose simplenatured preconditioning technique which explicitly involves the skew-symmetric part of the matrix. For a series of important applications the matrix skew-symmetric part may be get obtainable readily. For example, this is the case when, in FD-FE models, the central differences (the conventional Galerkin elements) has been used for approximate the first derivatives. Nevertheless in many cases obtaining of skew-symmetric part is not easy, and it might be a problem with applying the proposed technique. The preconditioners under considerations are in a certain sense related with the standard ILU- and SSORpreconditioners, but they don’t require a diagonal dominant and use only skew-symmetric part of the initial matrix. It is well done for strongly nonsymmetric linear equation system. It should be pointed out that since PTM-technique relates to alternatively triangular class of two-level stationary iterative schemes it admits affordable parallel and pipeline implementations. Numerical experiments have shown that PTM is good preconditioner for GMRES and not bad preconditioner for BiCG for Problems 3 and 4. So, it is possible to recommend PTM as preconditioner for GMRES and BiCG methods. Acknowledgements Supported by RFBR, grants N 12-01-00022 and N 12-01-31127.

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