Special iterative methods for solution of the steady Convection-Diffusion-Reaction equation with dominant convection

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1 Procedia Computer Science Volume 51, 015, Pages ICCS 015 International Conference On Computational Science Special iterative methods for solution of the steady Convection-Diffusion-Reaction equation with dominant convection L. A. Krukier 1, T.S.Martinova, B.L.Krukier 3, and O.A.Pichugina 4 1 Southern Federal University, Institute of MM and CS SFU, Rostov-on-Don, Russia krukier@sfedu.ru Southern Federal University, Institute of MM and CS SFU, Rostov-on-Don, Russia martynova@sfedu.ru 3 Southern Federal University, Institute of MM and CS SFU, Rostov-on-Don, Russia bk@sfedu.ru 4 Southern Federal University, Institute of MM and CS SFU, Rostov-on-Don, Russia pichugina@sfedu.ru Abstract Iterative methods based on skew-symmetric splitting of initial matrix, arising from central finite-difference approximation of steady convection-diffusion-reaction (CDR) equation in -D domain are considered. The property of obtained large sparse nonsymmetric linear system Au = f is investigated. A new class of triangular and product triangular skew-symmetric iterative methods is presented. Sufficient conditions of convergence for iterative methods of solution CDR with variable coefficient of reaction and dominant convection are obtained. The results of numerical experiments for the solution of a two-dimensional CDR equation are presented. The uniform grid, central differences for the first derivatives, natural ordering of points, Peclet numbers Pe =10 3, 10 4, 10 5, variable coefficient of reaction and different velocity coefficients have been used. Keywords: 1 Introduction The convection-diffusion-reaction (CDR) equation is the base for mathematical modeling in many fields of science and engineering. But up to now the main attention of researchers has This work was supported by RFBR, grant N a This work was supported by RFBR, grant N a This work was supported by RFBR, grants N and N a This work was supported by RFBR, grants N and N a Selection and peer-review under responsibility of the Scientific Programme Committee of ICCS 015 c The Authors. Published by Elsevier B.V. 139 doi: /j.procs

2 been connected with convection-diffusion (CD) problems and their numerical solution [11]. The most difficult problems for numerical solution of CD equation are [3]: 1. diffusion is quite small which means that the dimensionless parameter Pe > 10 3,. the field of velocity has stagnation points, Many different approaches have been proposed [15], [19], [3] to resolve the difficulties - exponential fitting, compact differences, upwinding, streamline diffusion [5], artificial viscosity and so on. Approximation of the first order derivatives in CD is the most interesting aspect of the solution for problem and very important work. It is well known [], [15] that using for approximation first order derivatives upwind schemes gives us linear equation systems with M- matrix [1], but matrix which can be obtained by using central FD schemes [6] is positive real. Each of these schemes have their own advantages and deficiencies which have been discussed in [3], [15]. When CRD equations investigate it is necessary to take into account the sign of reaction coefficient. If it is nonnegative then there is no problem with numerical solution, but if it is negative then the difficulties can arise. So if the negative coefficient reaction exists in CDR equation it means that after approximation it moves spectra of arising matrix in the left half part and matrix can lose property of being positive real. So, there is one more difficulty added to CDR equation in this case. 3. the coefficient of reaction is negative. Consider the convection-diffusion-reaction equation written in symmetric form [10] in bounded domain Ω = [0, 1] [0, 1] with boundary condition: ( 1 Pe ΔC + 1 u C x + (uc) x + v C y ) + (vc) y + αc = f(x, y), (1) C δω = C gr, () divũ =0, (3) where Pe is Peclet number, Ũ = {u, v} isthefieldofvelocityinω,c is unknown function, α is reaction coefficient, div Ũ = 0 (for incompressible medium), f is the right part of equation, δω is the boundary of Ω, C gr is the boundary value. Finite difference approximation of the equation The uniform grid Ω h with step h x = h y = h has been introduced in domain Ω. Introduce functions C(x i,y k )=C ik, x i = i 1 h, y k = k 1 h. All unknowns are calculated in the middle of the cell. The boundary conditions on Ω are interpolated on the boundary Ω h with a second order truncation error. The standard notation originating from [17] is used. The boundary conditions, with appropriate coefficients, are taken into account on the right-hand side of the difference equations. The central difference approximation of the first derivatives has been used. So, we obtain for (1) Pe Δ hc + 1 ( C i+1k C i 1k U ik h V ik C ik+1 C ik 1 h + V ik+1c ik+1 V ik 1 C ik 1 h + U i+1kc i+1k U i 1k C i 1k + (4) h ) + αc ik = f ik.

3 Here Δ h C is the difference analogue of Laplace operator. parts of equation by Peh.Then Transform (4), multiply both (4C ik C i+1k C i 1k C ik+1 C ik 1 )+ Peh U ik + U i 1k C i 1k + V ik + U ik+1 +αp eh C ik = Peh f ik [ Uik + U i+1k C i+1k C ik+1 V ] ik + U ik 1 C ik 1 + or where [( (4 + αp eh )C ik + 1+ Peh Ũik ( 1+ Peh ) Ṽik C ik+1 + Ũ ik = U ik + U i+1k ) ( C i+1k + 1 Peh ( 1 Peh Ṽik 1, Ũi 1k = U ik + U i 1k, Ũi 1k ) ] C ik 1 = f ik, ) C i 1k + (5) Ṽ ik = V ik + V ik+1 Thecoefficientsin(5)includethequantity, Ṽik 1 = V ik + V ik 1, f ik = Peh f ik. Re h = Peh/ (6) which was called cell Reynolds number or the skew-symmetry coefficient of the problem. 3 System of linear algebraic equation Using natural ordering of the unknowns, we transform (5) to the nonsymmetric linear system of equations ( Au = f, A = A Δ + A 1 + D, A 0 = ) ( 1 A + A T = A Δ + D = A T 0, A 1 = ) 1 A A T = A T 1, (7) where A is (N 1) (N 1) matrix, N = 1 h,u= {u 11, u 1,..., u N 1N 1 } T is the vector of solution, f= {f 11,f 1,..., f N 1N 1 } T is the vector of the right part. Matrix A can be naturally expressed [6] in the case of central difference approximation of the convective terms in (5) as a sum of symmetric positive definite matrix A Δ, skew-symmetric matrix A 1 and diagonal matrix D. A Δ is a difference analogue Δ h of operator Δ, describing a diffusion process, D is discrete analogue of the reaction term in the equation (1). A 1 is a difference analogue of the convective terms. Thus, linear system (7) with non-symmetric matrix A is constructed. If in (7) A 0 = A T 0 > 0, then matrix A is called positive real. 141

4 The linear system (7) is called strongly non-symmetric if A 0 / A 1 O(1), where is one of matrix norms. It can be easily verified that system (7) becomes strongly nonsymmetric for large values of Pe and α = 0. As a result we have A 0 =4, A 1 = Re h max i ( v 1i,j + v 1i,j 1 + v 1i,j + v 1i,j v i,j + v i+1,j + v i,j + v i 1,j )/, Theorem 3.1. Let equation (1) be approximated by finite difference scheme (5). system (7) is positive real if α 0. Then the Proof. The symmetric part A 0 of matrix A has the form A 0 = A Δ + D and doesn t have a definite sign in general case, but it is well known [17] that matrix A Δ is positive definite. So, if diagonal matrix D has nonnegative elements then A 0 will be positive definite as the sum of positive definite and nonnegative definite matrices. The last means that A 0 > 0ifα 0 and system (7) is positive real. It is well-known [15], [19] that using upwind scheme for equation (1) leads us to the system (7) with A being M-matrix [0], but in this case the obtained system won t be essentially nonsymmetric because matrix A has diagonal dominant. It is necessary to pay our attention [10] that the form in which we will approximate convection-diffusion equation plays a great role in successful numerical solution. Consider case when coefficient α<0. If Ω = [0, 1] [0, 1], boundary conditions are () and regular mesh is used, then eigenvalues and eigenvectors of L h = 1 Pe Δ h + α are well-known [18], [1]: λ mp (L h )= 4 ( Peh sin mπh +sin pπh ) + α, m =1,,...n 1; p =1,,...n 1, π Pe + α λ i 8 Peh + α, i =1,,...N, N =(n 1) (n 1). So, for α π Pe, difference operator for diffusion and reaction terms can lose the property of being positive real then from Hirsh theorem [13], its spectrum can move to the left half plane. Theorem 3.. Matrix (7), obtained from (5) is positive real, if 14 α conv π Pe.

5 4 Two-parameters skew-symmetric iterative solvers Besides important role in the mathematical modeling convection-diffusion-reaction equation is a good test for iterative methods. A lot of papers [1], [4], [3] have already described numerical experiments with CD or CDR equations for different parameters. Different basic iterative methods such as ILU [1], [3], [16], SOR [1], []have been used directly for solution of arising after approximation of CD or CDR equations by linear equation systems as well as preconditioners for CG or BiCG type s methods [14]. As it was shown in [1], ILU as a preconditioner for GMRES(0) and BiCGStab has been broken for large Re h, α =0 from (6) and natural ordering of the unknowns. We present a two parameters triangular [7] and product triangular iterative [] methods that use the skew-symmetric part of the matrix as an input and only require the matrix (7) to be positive real. Some ideas for using the splitting of skew-symmetric part of the matrix to solve linear equation systems arising after central difference approximation of first order terms in (5) have been firstly proposed in [6]. Let us approach (7) by considering the iterative method of the following form: B(ω) yn+1 y n + Ay n = f, n 0, (8) τ where f, y 0 H,H is an n-dimensional real Hilbert space, f is the right part of (7), A, B(ω) are matrices)in H, A is given by equation (7), B(ω) is invertible, y 0 is an initial guess, y k is the k-th approach, τ,ω > 0 are iterative parameters, u is the solution that we obtain, e k = y k u and r k = Ae k denote the error and the residual in the k-th iteration, respectively. Here it is important to note that B(ω) is in a certain sense a preconditioned matrix. In general, B(ω) is supposed to be nonsymmetric. Method (8) may be also represented as y n+1 = Gy n + τf, Consider the two ways of choosing matrix B. The first is G = B 1 (ω)(b(ω) τa). (9) B(ω) =B C + ω((1 + j)k L +(1 j)k U ), j = ±1, B c = B T c (10) and the second is B(ω) =(B C + ωk U )B 1 C (B C + ωk L ),B c = Bc T (11) where K L + K U = A 1,K L = KU T,B C = BC T. The matrices K L and K U represent strictly upper and lower triangular parts of a skew-symmetric matrix A 1 from (7) and matrix B C can be chosen arbitrarily, but has to be symmetric. These methods are called two-parameters triangular (TM) and product triangular (PTM) methods respectively. Matrix B is non-symmetric and can be represented as B = B 0 + B 1,B 0 = 1 ( B + B T ) = B0 T,B 1 = 1 ( B B T ) = B1 T. We find the symmetric and the skew-symmetric parts of matrix B for TM and PTM B 0 = B C + 1 ωj (K U K L ),j = ±1, B 1 = 1 ωa 1 (1) 143

6 B 0 = B C + ω K U B 1 C K L,B 1 = ωa 1. (13) The iteration matrix G from (9) for these methods is G = B 1 (B τa)=(b 0 + B 1 ) 1 (B 0 + B 1 τa 0 τa 1 ). (14) We consider the norm of iteration matrix G in (14). Let us require that matrices (10) and (11) are positive real and define matrices L 0T = B 0 1 ωa 0, (15) and L 0PT = B 0 ωa 0. (16) Using (1), (14) and (15) iterative matrix G T for TM can be represented as G T = ( B ωa 1) 1 ( B0 + 1 ωa 1 τa 0 τa 1 ) = = ( B 0 1 ωa ωa ωa 1) 1 ( B0 1 ωa ωa ωa 1 τa 0 τa 1 ) = = ( L 0T + 1 ωa) 1 ( L0T (τ 1 ω)a) and Introduce matrices and require for TM that P 0T = L 1 0T AL 1 0T, (17) P 0PT = L 1 0PT AL 1 0PT. (18) L 0T = B 0 1 ωa 0 = L T 0T > 0 (19) and for PTM Then L 0PT = B 0 ωa 0 = L T 0PT > 0 (0) G T = L 1/ ot (I + 1 ωp0t ) 1 (I (τ 1 ω)p0t )L 1/ ot = L 1 0T GP L 1 0T, G P =(I + 1 ωp0t ) 1 (I (τ 1 ω)p0t ). The last equality means that matrix L 0 generates energy norm G T L0T and G T L0T = (I + 1 ωp 0T ) 1 (I (τ 1 ω)p 0T ) (1) Lemma 4.1. Let C be a positive real matrix, α,β be positive numbers. If α<β α, α > 0. () Then (I + αc) 1 (I βc) < 1 (3) 144

7 Proof. First of all we point out that matrices (I + αc) 1 and (I βc) are commutative. Later we consider matrix T =(I + αc) 1 (I βc) and estimate its norm then and then Let So, if T Tv =sup ((I+αC) v 0 =sup 1 (I βc)v,(i+αc) 1 (I βc)v) v v 0 (v,v) = ((I βc)(i+αc) =sup 1 v,(i βc)(i+αc) 1 v) v 0 (v,v). u =(I + αc) 1 v T ((I βc)u,(i βc)u) =sup u 0 ((I+αC)u,(I+αC)u) =sup (u,u) β(cu,u)+β (Cu,Cu) u 0 (u,u)+α(cu,u)+α (Cu,Cu) = (Cu,u)+(α β)(cu,cu) =1 (α + β)inf u 0 (u,u)+α(cu,u)+α (Cu,Cu). α + β>0 α β 0 α>0 (Cu,u) 0 T < 1. and (3) fulfills. Inequalities (4) transform to (). Lemma 4.. [8] Let D = D T > 0 and A be positive real. Then (D + σa) 1 (D σa) < 1, D where σ>0 is parameter. Proof Let (4) First we note that T =(D + σa) 1 (D σa). and where T = D 1/ (I + σd 1/ AD 1/ ) 1 (I σd 1/ AD 1/ )D 1/ T D = T (5) T =(I + σm) 1 (I σm), M = C 1/ AC 1/ Then we obtain from (5) and Lemma 4.1 with α = β = σ result of Lemma 4.. We applied Lemma 4.1 to matrix G T in (1) and get following Theorem. 145

8 Problem No. v 1 v x y 1 3 x + y x y 4 sin πx πy cos πx Table 1: Velocity coefficients for test problems. Theorem 4.1. Let A in (7) be positive real. Then iterative method (8), (10) converges in H L0T if (19) fulfills and 0 <τ ω (6) Proof of this theorem consists of two step: -show that P 0T is positive real (Its the property of positive real matrix [18], if A is positive real, then C = QAQ T is positive real, too). So, from (17) P 0T is positive real. - insert in () α = 1 ω, β =(τ 1 ω) then we ve got (6). By similar lay out we can repeat for PTM just replace (1) on G T L0PT = (I + ωp0pt ) 1 (I (τ ω)p 0PT ) using (13), (14), (16), (18) and (0). Theorem 4.. Let A in (7) be positive real. Then iterative method (8), (11) converges in H L0PT if (0) fulfills and 0 <τ ω The proof of this theorem is the same of the previous one. 5 Numerical experiments In this section we present the results of numerical experiments in which the technique described above is used to solve nonsymmetric linear systems with α<0andα = 0. We compare the performance of SSOR [14] and PTM [] iterative methods to solve linear systems arising from the standard 5-point FD approximation of the steady convection diffusion-reaction problem (1) - (3) where F is chosen so that the solution of (1) is defined as ũ(x, y) =e xy sin πx sin πy. Equation (1) has been discretized by centered differences on a uniform grid with In the table 1 the used velocity coefficients of (1) are presented. Note that, for each model problem they are chosen to satisfy the constraint div v = v 1x + v y = 0 (which follows from the medium incompressibility for the problem ()). On the whole, in order to the test results to be comparable with those obtained in the other adjacent papers we take the analytical solution and the velocity coefficients similar to those in [3]. The initial guess in all runs was a zero vector and iterations were performed until / r m r , (7) 146

9 Pe Problem 1 Problem Problem 3 Problem 4 PTM SSOR PTM SSOR PTM SSOR PTM SSOR α = α = α = α = α = α = Table : Number of iterations for different α where r m is the residual vector, and represents the Euclidean norm. Checking and comparing iterative methods SSOR and PTM for different negative α (Table ) we show that methods are very good for α > 100. It means that matrix (7) is strongly diagonal dominant. This is connected with existence on main diagonal of elements a 1 =(4+αP eh ), which we obtain after approximation of coefficient of reaction. It includes numbers α and Re h, grows by module α and Re h. As we can see from the Table the number of iteration for α =0growswiththe increasing Peclet number. In contrast of this behavior of both iterative methods, the number of iteration decreases with grow of Peclet number and modula coefficient of reaction. 6 Conclusions The behavior of iterative methods to solve (7) which was obtained after approximation of CD (α =0)andCDR(α<0) equations is quite different (Table ). The case with α =0shows that matrix loses the property of diagonal dominance and the methods require more iterations as Re h increasing. Case with α 0forbignumbersα shows a very quick convergence of both methods for big numbers of Re h. 147

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