Direct linearization method for nonlinear PDE s and the related kernel RBFs

Transcription

1 Direct linearization method for nonlinear PDE s and the related kernel BFs W. Chen Department of Informatics, Uniersity of Oslo, P.O.Box 1080, Blindern, 0316 Oslo, Norway wenc@ifi.io.no Abstract The standard methodology handling nonlinear PDE s inoles the two steps: nmerical discretization to get a set of nonlinear algebraic eqations, and then the application of the Newton iteratie linearization techniqe or its ariants to sole the nonlinear algebraic systems. Here we present an alternatie strategy called direct linearization method (DLM). The DLM discretization algebraic eqations of nonlinear PDE s is simply linear rather than nonlinear. The basic idea behind the DLM is that we see a nonlinear term as a new independent systematic ariable and transfer a nonlinear PDE into a linear PDE with more than one independent ariable. It is stressed that the DLM strategy can be applied combining any existing nmerical discretization techniqes. The reslting linear discretization eqations can be either oer-posed or well-posed. In particlar, we also discss how to create proper radial basis fnctions in conjnction with the DLM. Key words: Nonlinear PDE s, Newton-type methods, direct linearization method, kernel radial basis fnction.

2 1. Introdction Althogh great endeaor has been deoted to nonlinear comptation and analysis, it seems ery difficlt to attack nonlinear problems directly. The linearization procedres sch as the Newton method and its ariants [1] are now commonly sed to transform a nonlinear system to a linear system in a point-wise approximate way so that the standard nmerical linear algebra approach can be employed for comptation and analysis. It is noted that the strategy of linearization often leads to a ery hge amont of compting effort and enconters great difficlty in nonlinear stability analysis. For instance, a nice initial gess soltion is key to garantee the conergence and reliability of the soltion, which, howeer, is often a danting task. Conseqently, a qestion is arosed: is iteration trly naoidable? The present stdy is a new qest among the others [2,3] to attack this problem in a different manner. The basic idea behind this stdy is that we see a nonlinear term as a new independent ariable and transfer a nonlinear PDE of an independent ariable into a linear PDE with more than one independent ariable. Then we can apply any standard nmerical discretization techniqe to analogize this linear PDE. To get the well-posed or oer-posed discretization formlations, we need to se staggered nodes a few times more of what the standard method reqires. It trns ot that the size of the reslting system matrix grows linearly proportional to the nmber of nonlinear terms. This methodology is called the direct linearization method (DLM). It is stressed that the DLM strategy can be applied with any existing nmerical discretization techniqes and essentially eliminates iteration linearization comptation. In conjnction with the DLM for nonlinear PDE s this paper also discsses three kernel BF strategies.

3 2. Direct linearization method In order to illstrate or idea clearly, let s consider the qadratic nonlinear eqation withot loss of generailty ( ) q( ) ( ) f ( x) p + =, (1) where p(), q() and () are linear differential operators, f(x) is inhomogeneos term. The mathematical description of the problem is complimented with Dirichlet and Nemann bondary conditions =, on Γ 1 (2a) q = n = q, on Γ 2, (2b) where n is the otward normal to the bondary, Γ=Γ 1 +Γ 2, and the pper bars indicate known bondary ales. The Newton-type methods of point-wise iteratie linearization are standard techniqe to sole nonlinear analog eqations of this PDE system. Instead, if we regard the qadratic nonlinear term as a new independent ariable, i.e., ( ) r( ) = p. (3) In this way, we rewrite nonlinear eqation (1) as a linear eqation with two independent ariables (x) and (x)

4 ( ) f ( x) + =. (4) Accordingly, we hae the Dirichlet bondary conditions for the new system independent ariable. It, howeer, is not easy to get Nemann bondary conditions of. We hae two approaches to sole this problem. The first is to se the particlar soltion method [4] or dal reciprocity method [5], which splits the original soltion into linear homogeneos soltion and nonlinear inhomogeneos particlar soltion. For the nonlinear particlar soltion part, we do not need to satisfy the bondary conditions at all. Therefore, we circment the Nemann bondary condition isse of the DLM. The second approach is that we simply replace Nemann bondary conditions of the DLM independent ariable with the goerning eqation. Since the original Nemann bondary conditions hae been satisfied by ariable, this approximation may not any introdce significant errors. The aboe steps clarify the basic idea behind the direct linearization method. Now we can se any nmerical discretization techniqe to analogize linear eqation (4). For instance, ariable and are approximately represented respectiely by a finite series approximation N ( x) = ( x) α k ϕ k k = 1, (5) N ( x) = ( x) β k φk k = 1, (6) where α and β are nknown expansion coefficients, φ and ϕ are the basis fnctions. N is the total nmber of bondary and inside-domain points. Now we need to discss how to choose basis fnctions. Is there any mtal constrains in choosing φ and ϕ once one of

5 them is determined? The similar isse also appears in handling a set of linear partial differential eqations with more than one independent ariable. We think that φ and ϕ shold satisfy somehow mtal constrains. Howeer, for this moment, this is still an open qestion and probably problem-dependent. In section 3, we will gie a brief discssion on this isse related to the radial basis fnction (BF) approach. Since we hae two independent ariables and and one eqation (4), we need to ealate 2N nknown coefficients. To get a well-posed or oer-posed discretization system of eqations, we need to discretize Eqs. (4), (2a,b), and bondary conditions of at least at 2N points. Namely, we reqire 2N staggered field points at bondary and inside domain. Then we hae 2N discretization eqations A α + β = (7a) 1 A2 b1 C α + β = (7b) 1 C2 b2 where A 1, A 2, C 1 and C 2 are interpolation matrices of order N. Note that formlations (7a,b) correspond to two sets of N staggered points across the problem domain and are a set of linear simltaneos eqations. Therefore, no iteratie linearization techniqe is reqired to sole the well-posed discretization eqations (7a,b). If we discretize eqation (4), (2a,b), and bondary conditions of at more than 2N points, we get an oer-posed linear system of eqations. Then a linear least sqare method shold be sed to sole the DLM discretization eqations. For more complicated copled nonlinear PDE systems, it is ery straightforward to apply the DLM.

6 3. Constrcting kernel BFs for the DLM In most practically significant cases, the nonlinear terms in the PDE system is a copling of seeral linear operators. The constrction of nonlinear algorithm shold consider this nonlinear featre. Following this idea, this section discsses the constrction of radial basis fnctions relating to the DLM. Chen and Tanaka [6,7] points ot an nderlying relationship between the BF approximation and Green identity. A kernel-bf strategy is proposed accordingly. By Green s second theorem, we hae soltion of Eqs. (1)-(2a,b) ( z) = w ( z, x) [ p( ) q( ) + f ( x) ] ( z, x) n dω + ( ) Γ Ω w z, x d Γ w n, (8) where w denotes the fndamental soltion of operator {}, x indicates sorce point. The aboe formla (8) sggests s that the exponential agmented kernel (EAK) BF can be created by 2m ( r x) = [ f ( x) + p( ) q( ) ] r w () r ω, (9), where m is an integral nmber and r 2m agmented term enhances the smoothness and ensres sfficient degree of differential continity since the fndamental soltion has a singlarity at origin. It is worth pointing ot here that we can se the nonsinglar general soltion instead of singlar fndamental soltion in arios kernel BFs presented in this paper. For the breity, we only mention the fndamental soltion relating to the kernel BF from now on.

7 In the BF approximation, the inflence coefficients are only-point dependent. Therefore, one factor decisie to the efficiency is to choose approximate inflence fnctions, which is termed as the BF normally. It is practically impossible for s to find accrate nonlinear inflence fnctions for complex problems. Simply remoing the nonlinear term in the BF (9), we hae an operational EAK BF 2m ( r x) = f ( x) r w (r) ω. (10), The aboe BF can be employed to analogize differential systems (1) and (2a,b) with the standard nmerical procedre. On the other hand, in terms of the DLM, we need to hae two BFs to respectiely approximate two independent ariables and in Eq. (4). = N k = 1 ( ) A k ψ r (11) k = N k = 1 ( ) B k ψ r (12) k Here a natral choice for ψ is the kernel BF for linear operator {} () r 2 r m ψ = w, (13) while ψ shold reflect nonlinear featre of operator p()q(), i.e., 2n ( r) = p( ) q( ) r w ( r) ψ. (14)

8 The key isse here is how to simplify the BF (14). One soltion is to se the fndamental soltion of operators p() and q(). Therefore, we hae 2n () r = r w () r w () r w () r ψ. (15) p q It is qite clear here that we se different operator-dependent kernel BFs to approximate independent ariable and, since they hae different physical meanings. From physical field iewpoint, we reason that the BF is in fact to ealate inflence coefficient of a sorce point in terms of inflence fnction of a physical problem. The BF (15) reflects to some extent physical backgrond of nonlinear terms. In other words, the present scheme is an intrinsically nonlinear nmerical discretization techniqe. The broad definition of the kernel BF inoles the se of nonsinglar high-order fndamental soltion and general soltion of operators and shape parameter. The second approach creating kernel BFs for and approximation is w m () r ψ =, (16) n n n () r = w () r w () r w () r ψ, (17) p q where m and n are the order of nonsinglar high-order fndamental soltions. This strategy is called high-order fndamental soltion kernel (HSK) BF. By analogy with sing the shape parameter in the MQ BF, we hae the third kernel BF creating strategy: 2 2 ( r c ) ψ, (18) = w +

9 () r = w ( r + c ) wp ( r + c ) wq ( r c ) ψ, (19) + where c is the shape parameter. Sbstitting 2 2 r + c into the BFs (16) and (17) instead of r is also an alternate approach. We simply name the present approach of the shape parameter kernel (SPK) BF. All complete fndamental soltions consist of essential and complementary elementary fnctions [8]. The standard singlar fndamental soltions sed in the BEM inole only the essential part. The complementary terms of the complete fndamental soltions are often nderstood the nonsinglar general soltion in terms of the bondary knot method [6,7]. The shape parameter c in the SPK and MQ BFs can be interpreted as the scaling parameter in the simplified form of the complete fndamental soltions and leads to infinite smoothness. More precisely, the SPK methodology constrcts the BF by sbstitting 2 2 r + c into the essential terms of the complete fndamental soltions [6,7] to compromise the complementary terms (general soltions). For instance, the reciprocal MQ is physically formed by this methodology sing the fndamental soltions of more than 3 dimensional Laplace operators. The MQ is based on the fndamental soltions of 1D Laplacian. The SPK thin plate splines are also presented in [6,7]. Comptational benefits sing the SPK depend on the tricky choose of the shape parameter, which is in agreement with the skillfl implementation of general fndamental soltions. The other possible approach embedding the complementary terms into the essential terms of the complete fndamental soltion is still open. In many cases, we may hae no fndamental soltions of operator p() and q(). The following scheme may be a cheap alternatie for creating BF for :

10 ψ 2s ( r) r p( ψ ) q( ψ ) =, (20) where s is an integral nmber. ψ can be nderstood approximate nonlinear inflence fnction, relatie to conentional linear inflence fnction. The essential philosophy behind the present BF for the DLM is that we shold think more the BF from a physical field point of iew than form a mathematical interpolation. Een if we se the standard single BF for soling nonlinear differential system (1) and (2a,b) withot relating to the DLM, the chosen BF shold be dependent on all differential operators. Based on the fndamental soltion of linear operator, the EAK BF is 2m [ ( ) q( w () r ) f ( x) ] r w () r ( r x) = p w ( r), + ω. (21) The HSK BF is m m m [ ( ) q( w () r ) f ( x) ] w () r ( r x) = p w ( r), + ω. (22) The SPK BF is ( r, x) = [ p( w ( r + c ) q( w ( r + c ) + f ( x) ] w ( r + c ) ω. (23) 4. Some remarks The basic ideas behind the preceding three kernel BF strategies for the DLM are also applicable to the polynomial approximation and general nonlinear data processing. All in

11 all, we shold nderstand nonlinear soler from system physical essence [8]. This stdy is still in a ery early stage. The practical nmerical experiments of this strategy will be proided sbseqently. eferences: 1. Ortega J.M., & heinboldt W.C., Iteratie Soltion of Nonlinear Eqations in Seeral Variables. Academic Press, New York (1970). 2. Chen W., Generalized linearization of nonlinear algebraic eqations: an innoatie approach, Co preprint: May Chen W., Generalized linearization in nonlinear modeling of data, Co preprint: Jly Atikinson K.E., The nmerical ealation of particlar soltions for Possion s eqation. IMA J. Nm. Anal., 5, (1985), Partridge P.W., Brebbia C.A. and Wrobel L.W., The Dal eciprocity Bondary Element Method. Compt. Mech. Pbl., Sothampton, (1992). 6. Chen W. and Tanaka M., elationship between bondary integral eqation and radial basis fnction. In Proc. of the 52th Symposim of Japan Society for Comptational Methods in Engineering (JASCOME) on BEM, Tokyo, Sept Chen W. and Tanaka M., New Insights into Bondary-only and Domain-type BF Methods. Int. J. Nonlinear Sci. & Nmer. Simlation. 1(3), (2000), Westphal Jr. T. and de Barchellos C.S., On general fndamental soltions of some linear elliptic differential operators. Engng. Anal. Bondary Elements, 17 (1996),