Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guang-wei Yuan Xu-deng Hang Laboratory of Computational Pysics, Institute of Applied Pysics and Computational Matematics, P.O.Box 9, Beiing, Cina Dedicated to te 7t birtday of Professor Lin Qun Abstract Tis paper discusses te accelerating iterative metods for solving te implicit sceme of nonlinear parabolic equations. Two new nonlinear iterative metods named by te implicit-explicit quasi-newton IEQN metod and te derivative free implicit-explicit quasi-newton DFIEQN metod are introduced, in wic te resulting linear equations from te linearization can preserve te parabolic caracteristics of te original partial differential equations. It is proved tat te iterative sequence of te iteration metod can converge to te solution of te implicit sceme quadratically. Moreover, compared wit te Jacobian Free Newton-Krylov JFNK metod, te DFIEQN metod as some advantages, e.g., its implementation is easy, and it gives a linear algebraic system wit an explicit coefficient matrix, so tat te linear inner iteration is not restricted to te Krylov metod. Computational results by te IEQN, DFIEQN, JFNK and Picard iteration metods are presented in confirmation of te teory and comparison of te performance of tese metods. Matematics subect classification: 65M6, 65M. Key words: Nonlinear parabolic equations, Difference sceme, Newton iterative metods.. Introduction For solving te implicit sceme of nonlinear parabolic problems from various applications, iterative metods are used wic adopt te inner-outer iteration mode. Te inner iteration is te linear iterative metods for te linearized systems, and te outer cycle is te nonlinear iterative metods wic will be discussed ere. To a great extent te outer nonlinear iteration determines te accuracy and efficiency of te total solution procedure. In te energy conservative equation of te radiation ydrodynamics, te diffusion coefficients and te source term are nonlinear wit respect to te temperature te temperatures of radiation, ion or electron. During te construction of te linearization procedure, te key point is to preserve te caracteristics of te original nonlinear parabolic equations so as to acieve ig efficient solution. In []-[5], it is pointed out tat te nonlinear convergence is tigtly relevant to te selection of time step and te precision of solution. Te efficient nonlinear iteration witin one time step can speed up te convergence of te iteration solution greatly. So it is essential to find ig efficient iterative metods in solving te nonlinear parabolic problems. Tere are at least tree reasons to prevent Newton metods applied in te nonlinear parabolic problems from some large scale scientific computations. Te first is tat te nonlinear Received Marc, 6. Supported by te Te National Basic Researc Program No. 5CB373 and te National Natural Science Foundation of Cina No.476, 6533.
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 43 iteration metods wit super-linear convergent order often ave local convergent region. In tis regard, a common approac is to reduce te time step to ensure te nonlinear iteration metod convergent. Actually, Newton metod is sensitive to te iterative initial value, and can be regarded as a measure of te nonlinearity. However, reducing frequently te time step will increase a lot of computational time. Te second reason is tat te Newton metod may cange te features of te original partial differential equations, wic makes te iteration ard to be convergent. A iteration metod of preserving te caracteristics of te original PDEs during te iteration process is more valuable tan te one tat possesses suc property only at te end of te iteration procedure. Te iteration metod of keeping te parabolic feature of te nonlinear parabolic equations not only ensures te efficiency of te computation, but also keeps te iteration solution to be positive see [7] for detail. Keeping te positivity in te iterative procedure is te foundation of te correct simulation of te pysical problem. Te tird reason is tat te Newton iteration sould form a Jacobian matrix, wic is often time-consuming, and is even impossible for some applications. For tis issue, some papers e.g. [4]-[6] suggest applying te JFNK Jacobian Free Newton-Krylov metod to deal wit suc problems. In tis paper, we pay attention to te last two reasons due to teir importance. Te main obective of tis paper is tat two new nonlinear iteration metods, called as te implicit-explicit quasi-newton IEQN metod and te derivative free implicit-explicit quasi-newton DFIEQN metod, are proposed. In tese metods we construct a iterative linearized difference sceme from te nonlinear implicit sceme, instead of simply applying te Newton metod or JFNK metod to te nonlinear algebraic system of equations. In oter words, te device of IEQN and DFIEQN metods are based on te nonlinear implicit sceme for te nonlinear parabolic equations, and not on te corresponding nonlinear algebraic system of equations. Moreover te performance of te DFIEQN metod is examined along wit some existing iteration metods including te semi-implicit metod SI, te fully implicit Picard metod FIP, fully implicit partial Newton metod FIPN and te JFNK Jacobian Free Newton-Krylov metod. Like JFNK metod, te DFIEQN metod is derivative free. But, unlike JFNK metod our DFIEQN metod as te advantage of FIP, i.e., its implementation is simple, and it gives a linear algebraic system wit an explicit coefficient matrix, so tat te inner iteration is not restricted to be cosen as te Krylov metod and it is more convenient and efficient to get a preconditioner. Moreover we will prove te DFIEQN metod is convergent quadratically, wile te SI, FIP and FIPN is convergent linearly see [7]. Te paper is organized as follows. Some nonlinear iterative metods are constructed in following section. Tese include te known semi-implicit SI metod, te fully implicit Picard FIP metod, and te fully implicit partial Newton FIPN metod. And ten we describe te construction of te implicit-explicit quasi-newton IEQN metod and te derivative free implicit-explicit quasi-newton DFIEQN metod. In te section 3 some assumptions and auxiliary lemmas are introduced, and te main convergence teorems are stated. In te section 4, we study te convergence property of te constructed nonlinear iteration metod, in particular we will prove te nd order convergence of te IEQN and DFIEQN metods. In te last section, numerical results are presented to sow te performance of tese metods.. Construction of te Iteration Sequences.. Te Problem and Some Notations To present te idea of te construction of te nonlinear iteration, te following one dimensional nonlinear parabolic problem is considered for simplicity ere u t Ax, t, uu x x = fx, t, u, Q T = { < x < l, < t T }. ux, = u x, x l. u, t = ul, t =, t T.3
44 G.W. YUAN AND X.D. HANG were Ax, t, u and fx, t, u are given functions of x, t, u, u x is a given function of x. For simplicity, we only consider te omogeneous boundary conditions.3 except in te section 5. Divide Q T by using parallel lines x = x =,,, J and t = t n n =,,, N, were x =, t n = n, and J = l,n = T, J and N are some positive integers, and are te space and time step. For n N denote te first order difference δu n + = u n + u n, =,,, J and te second order difference δ u n = δu n + δu n, =,, J. For a discrete function {u =,,, J} were u = u J = define some discrete norms as follows u = max u, δu = max δu + J J, J u = J u, δu = δu +. =.. Fully Implicit Sceme FIS A classical difference sceme for solving te problem..3 is te following implicit sceme u n were = A n+ δ A n+ = Ax + + A n+ = δ + f n+, J, n N.4 u = u x J.5 = J =, n N.6, tn+,, f n+ = fx, t n+,, and = un+ + + un+, x + = x + + x. Te basic properties of te implicit sceme for te nonlinear parabolic equations ave been studied in []..3. Semi-Implicit sceme SI In tis paper te semi-implicit sceme is referred to linearize te nonlinear equations.4 by constructing te diffusion coefficients and te source term by te last iterative values on te previous time level, i.e., u n = A n + δ A n + δ + f n, J, n N.7 wit te initial and boundary condition.5.6. Wen f = fu = u k, f n is often replaced by u n k or k k u n un k or some oter forms..4. Fully Implicit Picard Iteration FIP, usually called as te simple iteration For a fixed non-negative integer n n N, define a sequence of discrete functions { =,,, J} s =,, by te following way: { s+ =,,, J} is obtained by te solution of te following linear system of equations s+ u n = A n+ δ s+ A n+ δ s+ + f n+.
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 45 J, n N = u n J.9 s+ = s+ were s is te number of te iterations, and Wen f = fu = u k, A n+ = Ax + +, tn+, f n+ J =, n N., f n+ can be taken as s+ = fx, t n+,. k or k s+ k k. For te diffusion equation of non-divergence type, te convergence of FIP metod and te contraction of te iteration sequence are proved in [9]. For more general iterative difference scemes, it as been proved in [] tat te discrete solution converges to te solution of te diffusion equation if te time step and space step tend to zero..5. Fully Implicit Partial Newton Metod FIPN Te following FIPN metod differs from te FIP metod in te way of linearization of te nonlinear source term. It is defined by modifying te equation. into s+ u n = A n+ δ s+ A n+ δ s+ J, n N + f n+ + f n+ s+,. were f n+ = f x, t n+,. Te motivation to construct te metod is tat te diffusive term and te source term sould be managed separately for solving some practical problems. For example in te two-dimensional 3-temperature simulation of radiation ydrodynamics, te two nonlinear terms Auu x x and fu represent different pysical peculiarities. In many cases te energy excange term fu as a strong stiffness, and te implicit metod is required to resolve it, and a superlinear iterative metod is needed to make it converge nonlinearity as fast as possible. Wen Au is independent of u, te usual Newton metod is reduced to be.. Furtermore, in te Jacobian-Free metod f n+ fx, t n+, + εv f n+ ε were ε > is a small parameter..6. Implicit Explicit Quasi Newton Metod IEQN v can be approximated by Wen te Newton iteration metod and JFNK iteration metod are used to solve te nonlinear implicit sceme.4.6, people always consider te system.4.6 as a nonlinear algebraic system os equations Fu =, and ten define te iterative sequence { u : s =,, } by J s+ u u + F u =, were J is te Jacobian matrix for Newton metod or replacing derivatives wit difference quotients for JFNK metod. As we know, te Newton-Krylov metod and JFNK metod e.g., see [4]-[6] never manage te term Auu x x and fu separately. Wen solving te nonlinear
46 G.W. YUAN AND X.D. HANG algebraic system equations by te Newton metod, te parabolic properties of te original partial differential equations was seldom considered for te discretized and linearized system of equations. Following te argument metod used in tis paper, we can prove tat FIPN and FIP metods are of te first order convergence. Te convergence order is not iger tan one since te term Auu x x is linearized simply by A u s+ u x x. So ow to construct a nonlinear iteration metod wic is of superlinear convergence as well as preserving te parabolic property is very interesting. Te following IEQN metod will serve as an example. To sow clearly te mecanism of te IEQN metod, we consider te case fu for simplicity. To empasize te caracter of te metod, we omit te superscripts n + and subscripts if no confusion occurs. We still use δ to stand for te first order of difference and s stands for te number of iteration. If Au is linearized by one order Taylor expansion, tat is to replace A n+ A n+ s+ s+ u u n A n+, we get te following system = δ A + A s+ u u δ s+ u, J, n N. by A n+ + were = f x +, tn+,. Unfortunately, te system above is not linear, and can not be solved directly by linear solver. Under te assumption tat A + A s+ u u is always positive, we can prove te nd order convergence of te above iteration. But we can not ensure tat A + A s+ u u be always positive in practical computation, unless te time steps are cosen to be very small. Ten we propose te following IEQN metod: s+ u u n = δ Aδ s+ u + δ A s+ u u δ u, J, n N.. Tis metod can be obtained from FIP. by adding te δ A s+ u u δ u, wic is linear and one order difference wit respect to s+ u. Te resulting equations can preserve te parabolic property, and can be solved quickly since te quadratic convergence will be proved in te next section. Te IEQN metod result from te nonlinear implicit sceme.4 instead of te corresponding nonlinear algebraic system of equations, i.e., it is different from te Newton metod in tat it gives a direct approac to form Jacobian matrix. Furtermore it enligten us to propose te following derivative free implicit explicit quasi Newton metod DFIEQN, wic can be applied in some scientific and engineering computations..7. Derivative Free Implicit Explicit Quasi Newton Metod DFIEQN Now we describe te construction of DFIEQN iteration metod. In te IEQN metod. A we replace te derivative A wit te difference quotient { } A ε A ε = ε n+ A + ε n+ A, were ε n+ > are small parameters. Ten te DFIEQN iteration metod is constructed as
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 47 follows s+ u u n = δ Aδ s+ u + δ A ε s+ u u δ u, J, n N.3 wit te boundary condition.. Note tat te DFIEQN metod.3 as same principal part as te FIP., but tey differ in tat te FIP metod is of linear convergence wile DFIEQN is of quadratic convergence for te parameters ε cosen properly. Moreover a linear system of equations is formed for DFIEQN.3 wile it is not for JFNK. 3. Assumptions, Auxiliary Lemmas and Main Teorems 3.. Assumptions Introduce te assumptions: H A C R, and tere exists a constant σ > suc tat Av σ v R. H Te nonlinear implicit sceme.4.6 as one and only one solution { J, n N }, and tere exists a constant M > suc tat max n N δun+ M. H3 Let = u n un+. Assume and δ < c, were > and c > are small constants to be determined in te following section. 3.. Auxiliary Lemmas We need some lemmas see [] as follows: Lemma 3.. Te discrete Green formula Let u and v be te discrete function defined on {x =,,, J}, ten J J u v + v = u u v u v + u J v J. = = Lemma 3.. Te discrete Sobolev inequality For any discrete function u = {u =,,, J} J = l, te following assertions old. i For all ε >, tere are u ε δu + C ε u, were C is a constant depending on l, and independent of ε, and u ; ii If u = u J =, ten u l δu, u δu u ; iii Tere exist a constant C independent of and l, suc tat δu C u δ u + l u. In tis paper, C refers to a positive constant independent of,, and s te number of iteration, and may be different in different place. 3.3. Main Teorems Let be te solution of.4.6 wit f n+ for simplicity. Teorem. If te assumptions H H3 old and is small enoug, ten for te sequence { } defined by te IEQN metod. wit.9 and. tere old lim s+ + δ s+ =, lim δ s+ δ C,
4 G.W. YUAN AND X.D. HANG were =. Teorem. Assume H H3 old and is small enoug. If te sequence { by te DFIEQN metod.3 wit.9 and., and let ave i wen is satisfied ten tere olds ii wen is satisfied ten lim max J ε n+ lim δ s+ max J ε n+ lim = δ = ; δ s+ δ = O w, C. 4. Proof of Iteration Convergence = } is defined, ten we In tis section, we will prove te nd order convergence of IEQN and DFIEQN iteration metods, i.e., Teorem and. For simplicity te discrete indices and n + will be omitted if tere is no confusion. 4.. IEQN For { =,,, J} s =,, defined by te IEQN metod. wit.9. denote s+ w = δ Aδ s+ w =. From. and.4 we ave [ ] + δ A A + A s+ u u δu + A s+ u u δ u u, 4. were te term in [ ] at te rigt of 4. is equal to te following A u u + A s+ u u u u δu + A s+ u u u uδ u u = A w + A s+ w were te following abbreviations are used A A A A = A A A =,, were A = dr δu + A s+ w wδ w, A r A = + r A r [ d r A rr dr + dr, + ] r.
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 49 At te moment we assume A r + + C, A rr + + C. Here C and C are positive constants to be determined later. Multiplying 4. by s+ w, and summing up te products for =,, J, we obtain s+ w + σ δ s+ w J [ ] C w + s+ w δu + s+ w w δ w δ s+ w = σ δs+ w + C w 4 + s+ w + s+ w w δ w, were C is a constant dependent on M. So we ave C s+ w + σ δs+ w [ ] C δ w 4 + s+ w + w w C δ w 4 + s+ w δ s+ w w σ 4 δs+ w + C δ w 4 + s+ w. Ten, wen C, it can be deduced to s+ w + δ s+ w C δ w 4 + s+ w. Assume by induction C δ w s+ w + δ s+ w C <. Ten C δ w + s+ w < C + s+ w. It follows C δ s+ w <. So we conclude tat, if C δ w <, ten C δ w < s. Terefore s+ w + δ s+ w C δ w 4 C C δ w s+. Under te conditions of Teorem it is sowed te following results old s+ w + δ s+ δ s+ w w =, lim C. lim Te proof of Teorem is completed. 4.. DFIEQN δ w Now we give te proof of Teorem. Since it is similar to te argument of te above subsection, we only state te difference between tem. Now 4. sould be canged into s+ [ ] w = δ Aδ s+ w + δ A A + A s+ u u δu + A ε s+ u u δ u u [ ] +δ A ε A s+ u u δu, 4.
4 G.W. YUAN AND X.D. HANG were A ε = A ε n+ = = A ε A = dr A A ε = dr + ε n+ A A + r d r A r + r = ε n+ ε n+ A r + + [ d r A rr From te proof in 4. we can see tat as long as ten we can obtain Furtermore, if ten it follows + lim max J ε n+ lim max J ε n+ lim Terefore we ave proved te Teorem. δ s+ A + r A ε A r dr + r ε n+ r ε n+, + r ε n+ dr, + dr, ] + r ε n+. =, 4.3 δ =. 4.4 δ s+ δ = O, 4.5 5. Numerical Experiments C. 4.6 To compare te performance of te IEQN and DFIEQN wit oter metods, we give some numerical experiments on a model problem. Consider a model problem of te following form u t κ u κ u = S u 4, 5. x x y y were wit te boundary conditions for x, y [, ] [, ], t [, κ = z 3 u 3 ; z = z =, ux, y, =.d 5 4 u 6F = f, for x =, < y < ; u x =, for x =, < y < ; u x =, for x =, < y < u y =, for < x <, y = ; u y =, for < x <, y = 4 u + 6F =, for x =, < y < were F = z 3 u 3 u x
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 4 y u x = D u = y u = x D u = y x Figure : Te computational domain wit boundary conditions Figure : Te Robin boundary condition Te detail of te computational model is depicted in Figure. For te boundary conditions, te following discretization is adopted. For simplicity, we may consider a one dimensional case, te notions are listed in Figure. Since te boundary condition is valid on te boundary, ten it is equivalent to 4 u / u + 4 u 6 F / = f were u / denotes te value on te boundary. And we approximate te term 4 u / u by 4 u / u / 4 u x = / 4z 3 u 3 F / / wic is substituted into te discretized form of te boundary condition, and obtain / 4z 3 u 3 + / 6 F / = f 4 u. And replace te u / by u on te left side of te equation, we obtain te discretized approximation to F / as f 4 F / = u / + 4z 3 u 3 6 5. / Te equation 5. is integrated on a cell and discretized by finite volume metods, in wic te normal flux F n can be replaced by 5. if te side of te cell is on te boundary of Ω. In te present numerical experiments, we select S =.D; f =, z =
4 G.W. YUAN AND X.D. HANG and in eac of te above two subdomains, te mes size is 5 5. Te problem is solved by different nonlinear solvers different time steps. Tis model sows te conduction of eat flow wic is introduced from te left side, and flows out on te rigt side. Te process is sown by te following pictures. TIME=.5,DT=5.D-3 Z TIME=.5,DT=5.D-3 Z TIME=.5,DT=5.D-3 Z X Y ER X Y ER X Y ER 3 ER 4 Y 6 Figure 3. Energy distribution 6 X 4 3..6.4...6.4...6.4. 3 ER 4 Y 6 Figure 4. Energy distribution 6 X 4 3..6.4...6.4...6.4. 3 ER 4 Y 6 Figure 5. Energy distribution 6 X 4 3..6.4...6.4...6.4. GMRES is used as te linear solver, wit te preconditioner ILUTP,.d. For te JFNK test examples, FGMRES is used instead of te standard GMRES. Te program is written in FORTRAN, and run on a windows system. Two results of different time steps are presented below. History of nonlinear iterationt=.5,dt=5.d-3 5 History of nonlinear iterationt=.5,dt=.d-3, Number of iterations 5 FIP FIPN FIP FIPN JFNK IEQN DFIEQN Number of iterations 5 JFNK FIP FIPN IEQN/DFIEQN FIP FIPN JFNK IEQN DFIEQN 5 JFNKFAILED IEQN/DFIEQN...3.4.5 Pysical time Figure 6. Iterative istory of ex....3.4.5 Pysical time Figure 7. Iterative istory of ex. Te total time cost of eac iterative metod are listed in te following tablets. METHODS FIP FIPN JFNK IEQN DFIEQN IEQN ig precision TIME 5. 37. - 6.4 6.5. Table. Te time costs of te five metods DT= = 5.d 3 METHODS FIP FIPN JFNK IEQN DFIEQN IEQN ig precision TIME 3.6.. 6.9 63. 9. Table. Te time costs of te five metods DT= =.d 3
Acceleration Metods of Nonlinear Iteration for Nonlinear Parabolic Equations 43 Since te exact solution of te problem is unavailable, we solve te problem wit a iger precision. It is converged if te residual of eac nonlinear iteration is less tan.d. It is solved by IEQN, and te time consumption is listed in te last column of te above two tablets. And te solution is used as te approximate exact solution. Ten we can get te following results. Te relative error is te maximum norm of te relative error vectors. Te relative error of te four metodsdt=5.d-3-4 Relative error of te five metodsdt=.d-3.3-5 Te relative error.. FIP FIPN IEQN DFIEQN Relative error -6-7 FIP FIPN JFNK IEQN DFIEQN -...3.4.5 Te pysical time...3.4.5 Pysical time Figure. Relative error of ex. Figure 9. Relative error of ex. Te first experiment adopts time step 5.d 3. Te nonlinear convergence udgement is te residual norm is less tan.d 6. For tis time step, we notice tat JFNK failed, wile te oters succeeded. JFNK seems very sensitive to te time step. However, if JFNK converges, it converges fastest. And from te time and number of iterations, we find IEQN as well as DFIEQN te best. On most time, IEQN DFIEQN converges te fastest and costs te least time. For te second experiment, we use a smaller time step. Te nonlinear convergence udgement is te residual norm is less tan.d 6. In tis experiment, JFNK converges te fastest. IEQN in most place cost as many iterations as JFNK. But JFNK costs te most time. Tis is because te equation is nonlinear, and te discretization costs muc more time tan a matrix vector multiplication operation. For tis point, toug JFNK may be te fastest in convergence, it is not te fastest in time of solution. A tird result is tat DFIEQN and IEQN performs almost te same. In te two examples, we approximate te derivative of te diffusion coefficients by κ κu + ε κu, were ε =.d 6 u ε and te results sow tat it almost doesn t cange te convergence property of te IEQN metod. And if we solve te practical problem, wose diffusion coefficients are often provided by libraries, ten DFIEQN can be used instead of IEQN to get good performance. From te relative error of te problem Fig and Fig 9, we find for te large time step problem, te relative error of IEQN and DFIEQN are better in te former part of te simulation T <.5 and a little larger in te latter part were te IEQN only need two nonlinear iteration to converge. And in te small time step problem, te relative error of IEQN and DFIEQN is as good as FIP and FIPN, wile worse tan JFNK. But it takes still less time for IEQN to reac te same precision as JFNK. It takes about 9 seconds for te second example. In all, from te numerical examples, we find tat IEQN DFIEQN are sown to yield good convergence rate, stability and efficiency. References [] V.A. Mousseau, D.A. Knoll, W.J. Rider, Pysics-Based Preconditioning and te Newton Krylov
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