The Arithmetic Mean Solver in Lagged Diffusivity Method for Nonlinear Diffusion Equations

Transcription

1 American Journal of Computational and Applied Matematics 2012, 2(6): DOI: /j.ajcam Te Aritmetic Mean Solver in Lagged Diffusivity Metod for Nonlinear Diffusion Equations Emanuele Galligani Department of Matematics "G. Vitali", University of Modena and Reggio Emilia, Via Campi 213/b, I-41125, Modena, Italy Abstract T is paper deals wit te solution of nonlinear system arising from finite difference discretization of nonlinear diffusion convection equations by te lagged diffusivity functional iteration metod combined wit different inner iterative solvers. Te analysis of te wole procedure wit te splitting metods of te Aritmetic Mean (AM) and of te Alternating Group Explicit (AGE) as been developed. A comparison in terms of number of iterations as been done wit te BiCG-STAB algoritm. Some numerical experiments ave been carried out and tey seem to sow te effectiveness of te lagged diffusivity procedure wit te Aritmetic Mean metod as inner solver. Keywords Lagged Diffusivity, Aritmetic Mean Metod, Nonlinear Finite Difference Systems 1. Introduction We consider a nonlinear diffusion convection equation were te diffusion coefficient, denoted by σ, depends on te solution. Wen we use a finite difference discretization, tis elliptic equation supplemented by a suitable boundary condition, can be transcribed into a nonlinear system of algebraic equations. We wis to compute a solution of tis system of nonlinear equations wit a common iterative procedure in wic te nonlinear term, corresponding to te discretization of te diffusivity σ, may be evaluated at te previous iteration (see[22]). In literature, tis approac of nonlinearity lagging in te diffusivity term is denoted as Lagged Diffusivity Fixed Point Iteration or Lagged Diffusivity Functional Iteration. In Section 2, a model problem described by a nonlinear diffusion convection equation subject to omogeneous Diriclet boundary conditions is presented and a finite difference discretization is described. Ten, te lagged diffusivity procedure for te solution of tis nonlinear difference system is stated. Since, a purpose ere is to re-examine te lagged diffusivity procedure for solving te system of nonlinear difference equations of elliptic type in te context of Parallel Computing, te linear difference system tat arises at eac iteration of te lagged diffusivity procedure is solved wit te iterative splitting metods of te Aritmetic * Corresponding autor: emanuele.galligani@unimore.it (Emanuele Galligani) Publised online at ttp://journal.sapub.org/ajcam Copyrigt 2012 Scientific & Academic Publising. All Rigts Reserved Mean introduced in[20-21] and of te Alternating Group Explicit (AGE) introduced by Evans (see[1-3]). Tus, te outer iterates of te lagged diffusivity procedure are te approximate solutions of te linear systems computed wit an inner iterative solver; a criterion for acceptability of tese approximate solutions is given. A stopping rule for te lagged diffusivity procedure is also given. Section 3 is devoted to remind te Aritmetic Mean metod and AGE metod for te solution of linear systems wit block tridiagonal coefficient matrix. In Section 4, te convergence of te lagged diffusivity iteration metod is analysed under mild and reasonable assumptions imposed on te diffusivity σ using well known standard tecniques (see[16]). In Section 5, numerical experiments sow te beaviour of te inner-outer iterations of te procedure. In tis section a comparison of te lagged diffusivity iteration metod wit different iterative solvers is also presented. Te Aritmetic Mean and te AGE metods are compared, in terms of number of iterations, wit BiCG-STAB metod (see[23]). 2. Te Lagged Diffusivity Functional Iteration Metod Consider te model problem described by te nonlinear diffusion convection equation dddddd(σσ uu) + pp uu + qqqq = ff, (1) were u= u(x,y) is te density function at te point (x,y) of a diffusion medium R, σ =σ(u)>0 is te diffusion coefficient or diffusivity and is dependent on te solution u, q=q(x,y) 0 is te absorption term, te velocity vector p=(p 1, p 2 ) T is assumed to be constant and f(x,y) is a real valued sufficiently smoot function.

2 242 Emanuele Galligani: Te Aritmetic Mean Solver in Lagged Diffusivity Metod for Nonlinear Diffusion Equations In te boundary R o f R, equation (1) can be supplemented by a omogeneous Diriclet boundary condition of te form uu(xx, yy) = 0. (2) In te following, we suppose R to be a rectangular domain wit boundary R and we assume tat te functions σ, q and f satisfy te "smootness" conditions: (i) te function σ = σ(u) is continuous in u; te functions q(x,y) and f(x,y) are continuous in x,y respectively; (ii) tere exist two positive constants σmin and σmax suc tat 0 < σσ mmmmmm σσ(uu) σσ mmmmmm, uniformly in u; in addition, q(x,y) qmin 0; (iii) for fixed (x,y) R, te function σ(u) satisfies Lipscitz condition in u wit constant Γ (uniformly in x, y), Γ > 0. Te nonlinearity introduced by te u-dependence of te coefficient σ(u) requires tat, in general, te solution of equation (1) is approximated by numerical metods. We superimpose on R R a grid of points R R ; te set of te internal points R of te grid are te mes points (xx ii,yy jj ), for i = 1,..., N and j = 1,..., M, wit uniform mes size along x and y directions respectively, i.e. xx ii+1 = xx ii + and yy jj +1 = yy jj + for i = 0,..., N, j = 0,..., M. Tus, at te mes points of R R, xx ii,yy jj, for i = 0,..., N+1, j = 0,..., M+1, te solution uu(xx ii,yy jj ) is approximated by a grid function uu iiii defined on R R and satisfying te boundary condition (2) on R. In order to approximate partial derivatives in (1) we sall make use of difference quotients of grid functions. Te forward, backward and centered difference quotients wit respect to x and to y of te grid function uu iiii at te mes point (xx ii,yy jj ), are, respectively: xx uu iiii = uu ii+1jj uu iiii xx uu iiii = uu iiii uu ii 1jj, yy uu iiii = uu iiii +1 uu iiii,, yy uu iiii = uu iiii uu iiii 1, δδ xx uu iiii = 1 2 xxuu iiii + xx uu iiii, δδ yy uu iiii = 1 2 yy uu iiii + yy uu iiii, wile te centered second difference quotient wit respect to x and to y can be written δδ 2 xx uu iiii = xx xx uu iiii = xx xx uu iiii, δδ 2 yy uu iiii = yy yy uu iiii = yy yy uu iiii. Tis notation was introduced by Courant et al. in[4]. Providing a discretization error O( 2 ), te finite difference approximation of (1) in xx ii,yy jj, i = 1,..., N, j = 1,..., M, is given by xx σσ uu iiii xx uu iiii yy σσ uu iiii yy uu iiii + +pp 1 δδ xx uu iiii + pp 2 δδ yy uu iiii + qq xx ii,yy jj uu iiii = ff xx ii,yy jj, (3) By ordering in a row lexicograpic order te mes points P l = xx ii,yy jj, (i.e., l=(j-1) N+i wit j = 1,..., M, and i = 1,..., N), we can write te vector u of components uu iiii and te difference equations (3) as te nonlinear system FF(uu) AA(uu)uu ff = 00, (4) were te matrix A(u) is of order n = M N and as te block tridiagonal form; te M diagonal blocks are tridiagonal matrices of order N and te M-1 sub- and super- diagonal blocks are diagonal matrices of order N. Te five nonzero elements of A(u) corresponding to uu iiii 1, uu ii 1jj, uu iiii, uu ii+1jj and uu iiii+1 respectively, are (BB iiii + BB iiii ), (LL iiii + LL iiii ), (D ij +DD iiii ), (RR iiii + RR iiii ), and (TT iiii + TT iiii ), were LL iiii LL iiii (uu) = 1 σσ(uu 2 iiii ), RR iiii RR iiii (uu) = 1 σσ(uu 2 ii+1jj ), (5) BB iiii BB iiii (uu) = 1 2 iiii ), TT iiii TT iiii (uu) = 1 iiii+1), 2 and LL iiii = pp 1 2, RR iiii = pp 1 2, BB iiii = pp 2 2, TT iiii = pp 2 2, DD iiii DD iiii (uu) = BB iiii + LL iiii + RR iiii + TT iiii, (6) DD iiii = qq xx ii,yy jj. Te matrix A(u) is an irreducible matrix ([24, p. 18]). Providing tat te mes spacing is sufficiently small, i.e., < mmmmmm 2σσ mmmmmm pp 1, 2σσ mmmmmm pp 2, (7) te matrix A(u) is strict ly (wen qq xx ii,yy jj > 0 ) or irreducibly (wen qq xx ii,yy jj = 0) diagonally dominant ([24, p. 23]) and as positive diagonal elements, aa rrrr (uu) > 0, r = 1,..., n, and nonpositive off diagonal elements aa rrrr (uu) 0, rr ss, wit r,s = 1,..., n; terefore A(u) is an M-matrix ([24, p. 91] or[18, p. 110]). In te case of diffusion equation (pp = 00), te matrix A(u) is also symmetric; ten it is a symmetric and positive defin ite matrix ([24, p. 91]). Te vector f in (4) as components ff ll = ff xx ii,yy jj for i = 1,..., N, j = 1,..., M and l = (j-1) N+i. We remark tat wile te grid function uu iiii is defined on te wole mes region R R, te vector uu R nn represents te grid function uu iiii defined only on te interior mes points R. Here, we suppose tat a solution uu of te system (4) exists in B (see Section 4). For solving te nonlinear system (4) te easiest and maybe te most common metod is to lag te nonlinear term in (4) generating an iterative procedure denoted as Lagged Diffusivity Functional Iteration. Wit tis iterative procedure te nonlinear system (4) can be solved via a sequence of systems of linear equations. Specifically, given a sequence of positive numbers {εε νν } suc tat εε νν 0 as νν and an initial estimate uu (0 ) of te solution uu of te system (4), we generate a sequence of iterates uu (νν), νν = 0, 1,..., wit te following rule for te transition from a current iteration uu (νν) to te new iterate uu (νν+1) : Find an approximate solution uu (νν +1) of te linear system FF νν (uu) AA uu ( νν) uu ff = 00, (8) wit te criterion for acceptability of te solution on te norm FF νν uu (νν +1) εε νν+1. (9)

3 American Journal of Computational and Applied Matematics 2012, 2(6): Ten, te lagged diffusivity procedure is composed by an outer iteration tat generates te sequence uu (νν) and by an inner iterative solver of te linear system (8). Tis solver must be particularly well suited for implementation on parallel computers. Te termination criterion for te outer iteration is provided by te following stopping rule εε νν+1 εε, (10) were εε νν+1 decreases as εε νν +1 = 0.5 εε νν, νν = 1,2,, wit εε 1 = 0.1 FF uu(0) and εε is a prespecified tresold. 3. Iterative Parallel Solution of te Linear Systems In tis section, we remind te block form of te Alternating Group Explicit (AGE) and of te Aritmetic Mean metods for te solution of te linear system AAxx = bb, (11) wen te n n matrix A as te block tridiagonal form BB 1 CC 1 AA 2 BB 2 CC 2 AA = AA 3 BB 3 CC MM 13 CC MM 13 CC MM 13 CC MM 13 CC 3 AA MM 1 BB MM 1 CC MM 1 AA MM BB MM (12) Here, eac square block BB ii, i = 1,..., M, is a nonsingular N N matrix and te blocks AA ii and CC ii (i = 1,..., M; AA 1 = CC MM = 0) are square matrices of order N (n = N M). Te AGE metod consists in considering te following splitting of te matrix A AA = GG 1 + GG 2, were GG 1 and GG 2 are te following matrices BB 1 CC 1 AA 2 BB 2 BB 3 CC 3 GG 1 = AA 4 BB 4, BB MM 1 CC MM 1 AA MM BB MM BB 1 BB 2 CC 2 AA 3 BB 3 GG 2 =, BB MM 2 CC MM 2 AA MM 1 BB MM 1 BB MM wit BB ii = 1 BB 2 ii, i = 1,..., M. Tus, starting from a vector xx (0), te AGE metod generates a sequence of iterates xx (kk) as follows; for rr 0 and kk = 0, 1,..., until convergence: (GG 1 + rrii nn )xx ( kk+1 /2) = (rr II nn GG 2 )xx ( kk) + bb, (GG 2 + rrii nn )xx ( kk+1) = (rrii nn GG 1 )xx ( kk +1/2) + bb. (13) Here, II nn is te identity matrix of order n. Te AGE metod is convergent wen te matrices GG 1 and GG 2 are symmetric and positive definite wit rr 0. In tis case it is proved tat te optimal coice for r is rr = aaaa were 0 < aa λλ(gg 1 ) bb, 0 < aa λλ(gg 2 ) bb, and λλ ( GG ss ), ss = 1, 2, are te eigenvalues of te matrix GG ss (see, e.g.[1]). Furtermore, if te matrix A is irreducib ly (or strictly) diagonally dominant wit positive diagonal elements, aa iiii > 0, and nonpositive off diagonal elements, aa iiii 0, ii jj, and aa rr > max iiii, aa 1 ii nn or rr max iiii 2 1 ii nn, (14) 2 ten te AGE metod is convergent. Indeed, from te ypoteses we ave tat te matrix A is an M -matrix. Set PP ss = GG ss + rrii nn and QQ ss = rrii nn GG ss, ss = 1, 2. Te coice of r in (14) yields tat te matrices PP ss are strictly (or irreducib ly) diagonally dominant and ave positive diagonal elements and nonpositive off diagonal elements, ten PP ss are M-matrices wit PP 1 ss 0. Fro m te ypoteses on te matrix A, te matrices QQ ss are nonnegative QQ ss 0, ss = 1, 2. Tus, te iteration matrix T of te AGE metod is TT = PP 1 2 QQ 1 PP 1 1 QQ 2 0. Set PP 1 = 2rr PP 1 2 PP 1 1, (15) we ave PP 1 0 ( PP is an M -matrix) and TT = II nn PP 1 AA olds. Indeed, set AA = PP 1 QQ 2 = PP 2 QQ 1, a mult iplicative splitting metod can be written PP 1 xx ( kk +1/2) = (PP 1 AA)xx ( kk) + bb, PP 2 xx ( kk +1) = (PP 2 AA)xx ( kk +1/2) + bb. Ten, xx ( kk+1) = (II nn PP 1 2 AA)(II nn PP 1 1 AA)xx ( kk) +((II nn PP 1 2 AA)PP PP 1 2 ) bb, can be seen as a splitting metod (PP, QQ), i.e., AA = PP QQ, by setting PP 1 = (II nn PP 1 2 AA)PP PP 1 2 = PP 2 1(PP 1 + PP 2 AA)PP 1 1. Since PP ss = GG ss + rrii nn, ss = 1, 2, and AA = GG 1 + GG 2, ten we ave te expression (15) for PP 1. Now, te proof runs as tat of te Regular Splitting Teorem in[18, p. 119]. Te Aritmetic Mean metod uses te following two splittings of te matrix AA AA = HH 1 + KK 1, AA = HH 2 + KK 2, were HH 1 and HH 2 are te following matrices BB 1 CC 1 AA 2 BB 2 HH 1 = BB 3 CC 3 AA 4 BB 4, BB MM 1 CC MM 1 AA MM BB MM BB 1 BB 2 CC 2 AA 3 BB 3 HH 2 =, BB MM 2 CC MM 2 AA MM 1 BB MM 1 BB MM and KK 1 = AA HH 1, KK 2 = AA HH 2.

4 244 Emanuele Galligani: Te Aritmetic Mean Solver in Lagged Diffusivity Metod for Nonlinear Diffusion Equations We suppose M even. If M is odd, we can proceed in a similar way. Tus, starting from a vector xx (0), te metod of te Aritmetic Mean generates a sequence of iterates xx (kk ) as follows; for ρρ 0 and kk = 0, 1,..., until convergence: (HH 1 + ρρii nn )zz [1] = (ρρii nn KK 1 )xx ( kk) + bb, (HH 2 + ρρii nn )zz [2] = (ρρ II nn KK 2 )xx ( kk) + bb, (16) xx ( kk+1) = 1 2 (zz[ 1] + zz [ 2] ). In te paper[21], it is proved tat te block form of te Aritmetic Mean metod above described, is convergent wen te matrix A is: irreducibly (strictly) diagonally dominant wit positive diagonal entries and nonpositive off diagonal elements wit ρρ > 0 (ρρ 0); positive definite but not symmetric wit ρρ > ρρ were ρρ = max{ λλ mmmmmm (FF 1 ) λλ mmmmmm AA, λλ mmmmmm (FF 2 ) λλ mmmmmm AA } Here λλ(vv) denotes an eigenvalue of a matrix V and FF ss = HH ss HH TT ss KK ss KK TT ss, s=1,2, and AA is te symmetric positive definite matrix AA = AA + AA TT ; symmetric positive defin ite wit ρρ 0. At eac iteration k of te AGE metod, we ave to solve M /2 linear systems of order 2N, i = 1, 3, 5,..., M -1, BB ii + rrii NN CC ii xx ii AA ii +1 BB ii +1 + rrii NN xx = vv ii ii +1 vv, (17) ii +1 to obtain te vector xx ( kk+1 /2). Te solution of (17) can be seen as a block partitioned vector xx ( kk+1/2) kk = (xx TT kk , xx TT kk ,, xx TT MM ) TT, were eac block as N components. Here vv = (vv TT 1, vv TT 2,, vv TT MM ) TT is te rigt and side (r..s.) of te first equation of (13). Tese M / 2 systems can be solved simultaneously (in parallel). Ten, in order to obtain te new iterate xx ( kk+1) of te AGE metod, we ave to solve, in parallel, M / 2-1 systems as (17) wit i = 2,4, 6,..., M -2 and two linear systems of order N BB 1 + rrii NN xx 1 = vv 1, BB MM + rrii NN xx MM = vv MM, tat can be solved in parallel, as well. Te AGE metod as an intrinsic parallelism. In te case of te additive splitting metod of te Aritmetic Mean, at eac iteration k we ave to solve M-1 linear systems of order 2N, i = 1, 2, 3,..., M -1, BB ii + ρρii NN CC ii zz ii AA ii +1 BB ii +1 + ρρii NN zz = ww ii ii+1 ww, (18) ii +1 were zz = (zz TT 1,zz TT 2,, zz TT MM ) TT and ww = (ww TT 1, ww TT 2,, ww TT MM ) TT indicate te vector zz [1] (fo r i = 1, 3, 5,..., M -1) and te corresponding r..s. of te first equation of (16) or te vector zz [2 ] (for i = 2, 4, 6,..., M -2) and te corresponding r..s. of te second equation of (16). Furtermore, we ave to solve two linear systems of order N BB + ρρii 1 NN zz 1 = ww 1 BB + ρρii MM NN zz MM = ww MM were zz 1 and zz MM indicate te first and te last block of te vector zz [2] and ww 1, ww MM te corresponding r..s. of te second equation of (16). Tese systems can be solved in parallel. Te Aritmetic Mean metod introduces also an explicit parallelism in order to increase te degree of multiprogramming, tat is te number of processes tat can be executed simultaneously ([13, p. 87]). Wen te system (11) arises from te finite difference discretization of te problem (1)-(2), te diagonal blocks of A in (12) are tridiagonal wile te sub and superdiagonal blocks are diagonal. Tus, te systems (17) and (18) can be solved directly as in[9, 8] or iteratively, generating a two-stage iterative metod as in[12]. A direct solution of tese systems can be performed by cyclic reduction solvers ([17, p. 125],[15]) combined wit an approximate Scur complement metod ([17, p.123, p. 217]) or wit block Gaussian elimination. 4. Analysis of te Convergence of te Lagged Diffusivity Functional Iteration Metod In tis section, we prove te convergence of te sequence uu (νν) generated by te lagged diffusivity iteration metod for te solution of te system (4) under te smootness assumptions (i)-(iii). We define uu iiii and vv iiii, i = 0,..., N +1 and j = 0,..., M +1, te grid functions defined on R R and satisfying te Diriclet boundary condition on R. For grid functions uu iiii and vv iiii of tis type, te discrete ll 2 (RR ) inner product and norm are defined by te formulas NN MM uu, vv = 2 uu iiii vv iiii, NN MM uu = 2 ii =1 jj =1 ii =1 jj =1 uu iiii = uu, uu 1 2, respectively. We say tat te grid functions uu iiii defined on R R and vanising on R satisfy Property A if tey are uniformly bounded and ave uniformly bounded backward difference quotients xx uu iiii and yy uu iiii at eac mes point (xx ii,yy jj ) of R R. Te set of all grid functions wic satisfy Property A is denoted by B. Tus, B is te set of all grid functions uu iiii for wic tere exist two positive constants ρρ and ββ suc tat uu ρρ, (19) xx uu iiii ββ, yy uu iiii ββ, (20) Te constant ρρ is independent of ; also ββ is independent of but it depends on ff (see[16]). We assume tat te system (4), FF(uu) = 00, were FF(uu) AA(uu)uu ff, as at least one solution uu B.

5 American Journal of Computational and Applied Matematics 2012, 2(6): Suppose tat te system (4) arises from te discretization of problem (1)-(2) subject to te conditions (i)-(iii) wit qq qq mmmmmm > 0 and AA(uu) being an irreducible nonsingular M-matrix. If ΓΓ 2 ββ 2 < 1 2qq mmmmmm σσ mmmmmm for any uu B, ten te mapping FF(uu) is uniformly monotone in B. (See[16],[5] fo r a proof of t is result and e.g.,[19, p. 141] for te definition of uniform monotonicity). Te iterate uu (νν+1) of te lagged diffusivity functional iteration metod satisfies te system (8) wit te acceptability criterion (9), tat is uu (νν +1) is te solution of te linear diffusion convection equation wose diffusivity depends on te previous iterate uu (νν) wit inomogeneous term ff FF νν (uu (νν+1 ) ). We assume tat all te iterates uu (νν) satisfy Property A. Tus in particular, by inequality (20), te backward difference quotients of eac grid function are bounded and tey depend on te inomogeneous term; ten, we ave tat tere exist two constants ββ > 0 and ββ 0 > 0 suc tat (νν xx uu ) iiii ββ + εε νν ββ 0, yy uu (νν) iiii ββ + εε νν ββ 0, (21) for i = 1,..., N+1 and j = 1,..., M+1. Tus, we can state te following result concerning te convergence of te lagged diffusivity functional iteration metod. Teorem 1. Let uu be a solution of te nonlinear system (4) wit FF(uu) AA(uu)uu ff arising fro m te fin ite difference discretization of problem (1)-(2) subject to te conditions (i)-(iii) wit qq qq mmmmmm > 0 and AA(uu) being an irreducible nonsingular M-matrix. Assume tat te mapping FF(uu) is uniformly monotone in B. Suppose tat {εε νν } is a sequence of positive numbers suc tat εε νν 0 as νν. Let uu (00) B be arbitrary and let uu (νν +1) be te solution of FF νν (uu) = 00 satisfying te condition (9) wit FF νν (uu) as in (8). If all te vectors uu (νν ) belong to B and satisfy Property A wit (21) instead of (20), ten te sequence uu (νν) converges to uu. Proof. Te proof runs as tat in[6, Teor. 1] or in[5, Teor. 1, p. 33]. A more general result on te convergence of te lagged diffusivity functional iteration metod can be obtained by defining te mapping u=g(u) were GG(uu) = AA(uu) 1 ff, for all uu R nn. A solution of te system (4) is a fixed point of te mapping GG(uu). Since te smootness condition (iii) on σσ(uu), we can ave tat te matrix A(u) satisfies te Lipscitz-continuity condition for every bounded subset Ω of R nn wit a Lipscitz constant ΛΛ. Ten we can write, GG(uu) GG(vv) = AA(uu) 1 AA(vv) AA(uu) AA(vv) 1 ff. Tus, we ave for uu, vv Ω GG(uu) GG(vv) ΛΛ ff AA(uu) 1 AA(vv) 1 uu vv ΛΛ ββ 2 ff uu vv were ββ is a bound for AA(uu) 1 for any uu Ω. Here, indicates an arbitrary vector and matrix norm. Te last inequality assures tat te mapping GG(uu) satisfies a Lipscitz condition on a bounded subset Ω of R nn. 5. Numerical Experiments In tis section, we consider a numerical experimentation of te lagged diffusivity functional iteration metod for te solution of te nonlinear system (4) generated by te finite difference discretization above described, of te elliptic problem (1)-(2). Indeed, we ave to solve te system AA(uu)uu = ff. In tese experiments, te vector solution uu is prefixed and it is composed by te values of te prescribed function uu(xx, yy) = sin (ππππ)sin (ππππ) defined on te square [0, 1] [0, 1]. Te cosen functions for σσ(uu) are σσ1 σσ(uu) = 1 + uu; σσ2 σσ(uu) = aa bb + cc uu ; σσ3 σσ(uu) = 2(1 + 2uu 2uu 2 ); were, in te case of σσ2, we ave σσ2_1 for a=3, b=2, c=1; σσ2_2 for a=1.5, b=0.1, c=0.9 and σσ2_3 for a=1, b=0.01, c=0.99. Te vector ff is computed as ff ff = AA(uu )uu, were te matrix AA(uu ) of order n, as elements as in (5) and (6) wit N=M and nn = NN 2. In all te experiments we ave N=256 and pp 1 = pp 2 pp. At eac iteration νν, νν = 0,1,, of te lagged diffusivity procedure, we ave to solve te linear system of order n AA uu (νν) uu = ff, wit te splitting metod of te Aritmetic Mean or of te Alternating Group Explicit described in a previous section (see e.g.[9],[10] for an evaluation of te block form of te Aritmetic Mean metod on different parallel arcitectures and[7] for a description of te Fortran code implementing te metod). Tese metods are compared wit BiCG-STAB metod implemented as in[14, p. 50]. We call uu ( νν +1) te new iteration of te lagged diffusivity procedure, computed wit kk νν +1 iterations of te inner solver suc tat te inner residual FF νν uu ( νν +1) AA uu ( νν) uu ( νν +1) ff, satisfies te condition (9) FF νν uu (νν+1) εε νν +1, wit εε 1 = 0.1 FF uu(0) and εε νν+1 = 0.5 εε νν. (22) Here, indicates te Euclidean norm. Te vector FF uu(0) = AA uu ( 0) uu ( 0) ff is te in itial outer residual and its Euclidean norm is called rrrrrr0. Te initial vector uu (0) is taken as te null vector (uu (0) = 00) or as te vector e wose all te components are equal to 1 (uu (0) = ee). Te lagged diffusivity procedure as been implemented in a Fortran code wit macine precision and stops wen (10) olds, i.e., for νν = 0,1,,

6 246 Emanuele Galligani: Te Aritmetic Mean Solver in Lagged Diffusivity Metod for Nonlinear Diffusion Equations εε νν+1 εε, wit εε = We call νν te iteration of te lagged diffusivity procedure for wic condition (10) is satisfied. Tat is, te iterate uu (νν ) satisfies FF νν 1 uu (νν ) εε νν (εε νν > εε ), and we ave εε νν +1 = 0.5 εε νν and εε νν +1 εε. In te tables, we report te number of iterations νν and, in brackets, te total number of iterations of te inner solver kk TT, i.e., kk TT = kk νν. νν =1 In te tables, we also report te discrete ll 2 (RR ) norm of te error, eeeeee = uu uu νν, te Euclidean norm of te outer residual rrrrrr = FF uu(νν ) = AA uu ( νν ) uu ( νν ) ff, and rrrrrr0. Te symbol close to te value of rrrrrr indicates tat te beaviour of te norm of te outer residual FF νν uu (νν+1 ) is not monotone decreasing. Te writing max it. indicates tat at a certain iteration νν, te ma ximu m number of iterations of te inner solver as been reaced. Te maximum number of inner iterations is set equal to Te writing n.c. indicates tat at a certain iteration νν, te condition (7) is not satisfied. Te symbol " " close to te number of inner and outer iterations denotes tat at a certain iteration νν, te condition (7) is not satisfied and, in tese cases, te lagged diffusivity iteration metod generates te iteration by performing a prefixed number (equal to 20) of iterations of te inner iterative solver. Te writing 1.71( 8) indicates In te tables, we indicate wit lag-am, lag-age and lag-b, te lagged diffusivity functional iteration metod wit te Aritmetic Mean, te Alternating Group Explicit and te BiCG-STA B metod, respectively, as inner solver. Furtermore, we observe tat, since εε νν+1 decreases, for νν increasing, as (22) and te lagged diffusivity functional iteration metod stops at te iteration νν wen te criterion for εε νν +1 in (10) is satisfied, we ave εε νν +1 = 1 2 εε νν = 1 2 εε νν 1 = 1 εε 2 νν 1 εε, were we set εε 1 = 0.1 FF uu(0). Ten, νν > log 2 εε 1. εε Indeed, in te experiments we obtain νν = log 2 εε 1. εε νν Table 1. Result s for different values of p for σσ1 N=256 σσ(uu) = σσ1 q=0 uu (0) = (478) 29 (538) 29 (1399) (547) 28 (621) 28 (1189) (720) 28 (851) 28 (1118) (1062) 27 (1243) 27 (1175)* (2267) 26 (2709) 26 (1022)* (5131) 25 (6088) 25 (946) (31110) 23 (35322) 23 (1202) (55159) 22 (64165) 22 (1357)* Ta ble 2. Result s for different values of p for σσ2_1 σσ(uu) N=256 q=0 uu = σσ2_1 (0) = (441) 29 (470) 29 (1189) (524) 28 (565) 28 (1110) (681) 28 (757) 28 (1108)* (972) 27 (1088) 27 (1050) (1965) 26 (2208) 26 (1034) (4232) 25 (4857) 25 (864) (24409) 23 (30782) 23 (1149) (60546) 22 (79276) 22 (1581) Ta ble 3. Result s for different values of p and uu (0) for σσ2_2 N=256 σσ(uu) = σσ2_2 q=0 uu (0 ) = n.c. n.c. max it.* n.c. n.c. max it.* (21137) 26 (70137) 26 (2767) (56063) 26 (343861) 26 (3907) (80638) max it. 26 (4449) uu (0) = ee (1272) 35 (4657) max it.* (1616) 35 (5671) max it.* (2154) 35 (7633) max it* (3120) 35 (11226) max it* (6153) 35 (26619) 35 (2782) (12048) 35 (57421) 35 (2724) (46793) 35 (342113) 35 (4235) (72524) max it. 35 (4842)

7 American Journal of Computational and Applied Matematics 2012, 2(6): Ta ble 4. Result s for different values of p and uu (0) for σσ2_3 N=256 σσ(uu) = σσ2_3 q=0 uu (0) = n.c. n.c. max it.* n.c. n.c. max it.* (23198) max it. max it.* (53247) max it. 30 (10173) (102915) max it. 30 (13616) uu (0) = ee (2486)* n.c. max it.* (2516) n.c. max it (3108) 38 (68052) max it.* (4255) 38 (108203) max it.* (7153) max it. max it.* (11826) max it. max it.* (40302) max it. max it.* (93196) max it. max it.* Ta ble 5. Result s for different values of p for σσ3 N=25 6 σσ(uu) = σσ3 q=0 uu (0) = (765) 29 (818) 29 (1161)* (953) 28 (1026) 28 (1081) (1253) 28 (1341) 28 (1039) * (1974) 27 (2118) 27 (1016) (4339) 26 (4683) 26 (924) (9725) 25 (10525) 25 (1054) (46400) 24 (53706) 24 (1565) (61128) 23 (71699) max it.* Ta ble 6. Results for different values of q N=256 σσ(uu) = σσ1 uu (0) = 00 q νν (kk TT ) rrrrrr0 p = (14136) 24 (18027) 24 (1069) (13217) 24 (16868) 24 (998) (8814) 25 (10794) 25 (600) (2118) 27 (2383) 27 (211) p = (1062) 27 (1243) 27 (1175) (1061) 27 (1245) 27 (1185)* (1039) 27 (1232) 27 (1124)* (776) 28 (928) 28 (860) (267) 31 (290) 31 (128) N=256 σσ(uu) = σσ2_3 p=100 uu (0 ) = ee 0 38 (7153) max it. max it.* (6468) 38 (214471) max it.* (3664) 38 (97128) max it.* (1117) 38 (15122) max it.* Ta ble 7. Some results for te error and te residuals N=256 σσ(uu) = σσ1 q=0 uu (0) = 00 metod eeeeee rrrrrr rrrrrrrr p=1 lag-am 1.71(-8) 1.54(-4) lag-age 1.71(-8) 1.62(-4) lag-b 7.78(-9) 1.90(-4) p=50 lag-am 1.49(-9) 1.69(-4) lag-age 9.71(-10) 1.71(-4) lag-b 1.62(-10) 1.70(-4) p=100 lag-am 5.03(-10) 1.69(-4) lag-age 3.12(-10) 1.69(-4) lag-b 4.30(-11) 4.90(-5) p=300 lag-am 6.89(-11) 1.19(-4) lag-age 4.24(-11) 1.20(-4) lag-b 1.42(-11) 1.21(-4) 6. Conclusions Fro m te numerical experiments te following conclusions can be drawn: te outer residual rrrrrr as te same order of εε and te error eeeeee in te discrete ll 2 (RR ) norm as, in worst cases, order εε; te AM metod gives better results wen te ratio between te maximum value of σσ and te smallest component of pp is small, tat is te coefficient matrix of te linear system is strongly asymmetric (see[20]) or te deviation from asymmetry is decreasing. We define as te deviation from asymmetry of a matrix te difference between te Frobenius norms of te symmetric and nonsymmetric parts of te matrix. Furtermore, we can observe te same beaviour of te AGE metod wit te one of te AM metod, respect to te deviation from asymmetry of te coefficient matrix tat occurs at eac step of te lagged diffusivity procedure; te lagged diffusivity functional iteration metod combined wit te AM or te AGE metod breaks down wen te coefficient matrix, at a certain iteration νν, is not an M -matrix (n.c.). In some cases, te AGE inner solver, requires a large number of inner iterations especially for nearly symmetric matrices; te beaviour of te residual rrrrrr wen we use AM or AGE metod as inner solver, is always monotone, except in one case were te lagged diffusivity procedure breaks down (te coefficient matrix is not an M-matrix) but it as been possible to force te convergence by running a few number of iterations of te inner iterative solver. Te nonmonotonicity of te residual appens at tese "forced" iterations. Tis tecnique of forcing convergence is successful wen condition (7) is "almost satisfied". te lagged diffusivity functional iteration metod combined wit te BiCG-STAB metod, wen it does not break down, requires a number of inner iterations tat is not large and seems to be independent from te deviation from

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