A MANIFOLD-BASED EXPONENTIALLY CONVERGENT ALGORITHM FOR SOLVING NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 Journal of Marine Science and Technology Vol. No. pp. -9 () DOI:.69/JMST--- A MANIOLD-ASED EXPONENTIALLY CONVERGENT ALGORITHM OR SOLVING NON-LINEAR PARTIAL DIERENTIAL EQUATIONS Chein-Shan Liu Key words: non-linear algebraic equations (NAEs) non-linear partial differential equations manifold-based eponentially convergent algorithm (MECA) fictitious time integration method (TIM). ASTRACT or solving a non-linear system of algebraic equations of the type: i ( j ) i j n a Newton-like algorithm is still the most popular one; however it had some drawbacks as being locally convergent sensitive to initial guess and time consumption in finding the inversion of the Jacobian matri i / j. ased-on a manifold defined in the space of ( i t) we can derive a system of non-linear Ordinary Differential Equations (ODEs) in terms of the fictitious time-like variable t and the residual error is eponentially decreased to zero along the path of (t) by solving the resultant ODEs. We apply it to solve D non-linear PDEs and the vector-form of the matri-type non-linear algebraic equations (NAEs) is derived. Several numerical eamples of non-linear PDEs show the efficiency and accuracy of the present algorithm. A scalar equation is derived to find the adjustive fictitious time stepsize such that the irregular bursts appeared in the residual error curve can be overcome. We propose a future direction to construct a really eponentially convergent algorithm according to a manifold setting. I. INTRODUCTION Paper submitted /3/; accepted //. Author for correspondence: Chein-Shan Liu ( liucs@ntu.edu.tw). Department of Civil Engineering National Taiwan University Taipei Taiwan R.O.C. In many practical non-linear engineering problems governed by partial differential equations the methods such as the finite element method boundary element method finite volume method the meshless method etc. eventually led to a system of non-linear algebraic equations (NAEs). Many numerical methods used in the computational mechanics such as demonstrated by Zhu et al. [3] Atluri and Zhu [6] Atluri [] Atluri and Shen [5] and Atluri et al. [3] were always led to the solution of a system of linear algebraic equations for a linear problem and of an NAEs system for a non-linear problem. Over the past forty years two important contributions have been made towards the numerical solutions of NAEs. One of the methods has been called the predictor-corrector or pseudo-arclength continuation method. This method has its historical roots in the embedding and incremental loading methods which have been successfully used for several decades by engineers to improve the convergence properties when an adequate starting value for an iterative method is not available. Another is the so-called simplical or piecewise linear method. The monographs by Allgower and Georg [] and Deuflhard [] are devoted to the continuation methods for solving NAEs. Liu and Atluri [3] have employed the technique of ictitious Time Integration Method (TIM) to solve a large system of NAEs and showed that high performance can be achieved by using the TIM. More recently Liu [7] has used the TIM technique to solve the non-linear complementarity problems. Then Liu [8 9] has used the TIM to solve the boundary value problems of elliptic type partial differential equations. Liu and Atluri [] also employed this technique of TIM to solve mied-complementarity problems and optimization problems. Liu and Atluri [5] using the technique of TIM solved the inverse Sturm-Liouville problem under specified eigenvalues. Liu [] has utilized a simple TIM to compute both the forward and backward in time problems of urgers equation and has found that the TIM is robust against the disturbance of noise. Liu and Chang [8] have used the TIM to solve ill-posed linear problems. or its numerical implementation being quite simple the TIM was also used in other problems like as Liu [ ] Liu and Atluri [6] Chang and Liu [8] Chi et al. [] and Tsai et al. [3]. In spite of its success the TIM is local convergence and needs to determine viscous damping coefficients for different equations in one problem. Liu and Atluri [7] were the first to

2 Journal of Marine Science and Technology Vol. No. () make a breakthrough by deriving a residual-norm based algorithm which can circumvent the drawbacks of TIM and Newton s algorithm; also refer Atluri et al. []. In this paper we introduce a novel continuation method of Manifold-ased Eponentially Convergent Algorithm (MECA) which can be easily implemented to solve nonlinear partial differential equations after some suitable discretizations by using the finite difference method or epansion by the radial basis function []. To remedy the shortcoming of vector homotopy method as initiated by Davidenko [] Liu et al. [9] have proposed a scalar homotopy method and Ku et al. [5] combined this idea with the eponentially decayed scalar homotopy function developing a Manifold- ased Eponentially Convergent Algorithm (MECA). In this paper we will point out the limitation of MECA and eplain why it may stuck and after that the algorithm cannot proceed to find solution. II. THEORETICAL ASIS or the following non-linear algebraic equations (NAEs) in a vector form: ( ) () we will propose a numerical algorithm to solve it based on a space-time manifold.. Motivation To motivate the present approach let us consider an uncoupled algebraic system: y. () The above two equations can be combined into a single one: [( ) ( ) ]. + y (3) We can see that it is a circle in the spatial-plane ( y) with a center () but with a zero radius. Now we introduce an etra variable t as a fictitious timelike variable and let and y be functions of t. We consider the following equation defined in the space-time domain by αt αt e e [( ) + ( y ) ] C () where α > and C is determined by the initial values of () and y() y by C [( ) + ( y ) ]. Eq. () can be written as ( ) ( y ) Ce αt + (5) which is a manifold in the space-time domain ( y t) at each cross-section of a fied t it being a circle with center () and with a radius Ce αt. We can construct the following ODEs: α( ) y α( y ) (6) such that the path of ((t) y(t)) generated from the above ODEs is located on the manifold defined by Eq. (5). Solving Eq. (6) we have αt t () [ ] e + yt () [ y ] e +. (7) αt Inserting them into Eq. () we indeed can prove that ((t) y(t)) is located on the manifold and ((t) y(t)) fast tends to the solution ().. A Setting ased on Manifold rom the above idea of space-time manifold for Eq. () we can consider h( t): Q( t) ( ) C (8) where Q(t) > is a given function of t monotonically increasing with t and with Q() and C is determined by the initial condition () with C ( ). (9) Usually C > and C when the initial value is just the root of Eq. (). However it is rare of this lucky case. When C > and Q > the manifold defined by Eq. (8) is continuous and thus the differential operation being taken on the manifold makes sense. or the requirement of consistency condition by taking the differential of Eq. (8) with respect to t and considering (t) we have () ( ) ()( ) Qt Qt + () where is the Jacobian matri with its ij-component given by ij i / j. Eq. () cannot uniquely determine the governing equation of ; however we suppose that h λ λqt () ()

3 C.-S. Liu: A Manifold-ased Algorithm 3 where λ is to be determined. Inserting Eq. () into Eq. () we can solve for λ by λ Qt () Q ( t). () Thus we obtain an evolution equation for defined by the following ODEs: III. NUMERICAL METHODS. Adjusting the ictitious Time Step In order to keep on the manifold defined by Eq. (8) we can consider the evolution of along the path (t) by qt () A (9) qt () (3) where A:. () where Qt () qt ():. Qt ( ) () There are many ways to choose a suitable function of Q(t); however we can let Hence we have Qt () ν qt () < m. Qt ( ) ( + t) m ν Qt + m m () ep [( t) ]. Therefore we can derive the following equation:. (5) (6) ν (7) ( + t) m It is not difficult to prove that the orbit (t) with () solved from Eq. (7) is located on the space-time manifold defined by Eq. (8) with Q(t) given by Eq. (6). Hence we have an eponentially convergent property in solving the NAEs in Eq. (): Suppose that we simply use the Euler forward scheme to integrate Eq. (9) which yields ( t+ t) () t q() t t A. () Taking the square-norms of both the sides and using Eq. (8) we can obtain C C C ( A) qt ( ) t Qt ( + t) Qt () Qt () C ( qt ( ) t) A. + Qt () Thus we have the following scalar equation to solve t: where Qt () f( t): a( t) b t + Qt ( + t) A : ( ) a q t () (3) () b: q( t). (5) C ( ). (8) Qt () When t is increasing the above equation enforces the residual error ( ) tending to zero eponentially and meanwhile the solution of Eq. () is obtained approimately. However it is a great challenge by developing a suitable numerical integrator for Eq. (7) such that the orbit of can really retain on the manifold. Obviously t is a root of Eq. (3); however we prefer to search another one with t >. On the other hand we note that ν in Eq. (6) cannot be too large say ν > ; otherwise the term Q(t)/Q(t + t) in Eq. (3) can be very near to zero and thus makes Eq. (3) having no real solution because of ( b) a q A <

4 Journal of Marine Science and Technology Vol. No. () which is due to ( A) < A. (6) We can apply the TIM to find the solution of Eq. (3): µ f( ). (7) ( + t) γ variables K and the vectorial algebraic equations K by Do i n i j i j Do j n K n ( i ) + j K K u. (3) When t is solved we can use the following iteration to calculate (t + t): ( t+ t) () t q() t t. (8) The above algorithm is indeed a very powerful numerical method with eponentially convergent speed and the orbit of (t) being retained on the manifold. Thus we may call the present algorithm a Manifold-ased Eponentially- Convergent Algorithm (MECA).. A Matri Type NAEs Sometimes we may encounter the NAEs generated from the discretization of PDE with a matri type. In this situation it is not so straightforward to write its counterpart as being a vector-form NAEs. Let us consider uy ( ) yuu ( u )( y ) Ω (9) y uy ( ) Hy ( )( y ) Γ (3) where is the Laplacian operator Γ is the boundary of a problem domain Ω : [a a ] [b b ] and and H are given functions. y a standard finite difference applied to Eq. (9) one has a system of NAEs of matri type: i j [ u i+ j ui j+ ui j] + [ u i j+ ui j+ ui j ] ( ) ( y) u u u u y i n j n. i+ j i j i j+ i j i yj ui j (3) Here we divide the rectangle Ω by a uniform grid with (a a )/(n + ) and y (b b )/(n + ) being the uniform spatial grid lengths in the - and y-direction and u i j : u( i y j ) be a numerical value of u at the grid point ( i y j ) Ω. Let K n (i ) + j and with i running from to n and j running from to n we can respectively set the vectorial At the same time the components of the Jacobian matri are constructed by Do i n Do j n K n ( i ) + j L n ( i ) + j L n ( i ) + j L n ( i ) + j 3 L n ( i ) + j+ L5 ni+ j ( ) ui+ j ui j ui j+ ui j + u i yj ui j y KL ( y) ui+ j ui j ui j+ ui j + u y i yj ui j y y KL ( ) ( y) KL 3 u u u u y i+ j i j i j+ i j u i yj ui j ( y) ui+ j ui j ui j+ ui j u y i yj ui j y y KL ( ) KL 5 ui+ j ui j ui j+ ui j u. i yj ui j y (33)

5 C.-S. Liu: A Manifold-ased Algorithm 5 In above u u and uy denote respectively the partial differentials of the function ( y u u u y ) with respect to u u and u y. When the above quantities are available we can apply the vector-form MECA to solve the non-linear PDE in Eq. (9). IV. NUMERICAL TESTS O MECA In this section we apply the new method of MECA to solve some non-linear PDEs.. Eample We consider a non-linear heat conduction equation: 8 6 α u α u + α u + u + h t (3) t ( ) '( ) ( ) ( ) ( 3) ( ) 7( 3) t t h t e ( 3) e (35).7.6 with a closed-form solution u( t) ( 3) e t. y applying the MECA to solve the above equation in the domain of and t we fi n n 3 ν 5 and m. We apply the group-preserving scheme (GPS) developed by Liu [6] to integrate the resultant ODEs with a fictitious time stepsize t.5. In ig. we show the residual errors with respect to the number of steps up to. The absolute errors of numerical solution are plotted in ig. which reveal an accurate numerical result with the maimum error being Eample One famous mesh-less numerical method to solve the non-linear PDE of elliptic type is the radial basis function (R) method which epands the solution u by Error of u ig.. or eample : showing the convergence speed of residual errors and displaying the numerical error. t n uy ( ) aφ (36) k where a k are the epansion coefficients to be determined and φ k is a set of Rs for eample N 3/ φk ( rk + c ) N N φk rk ln rk N r k φk ep a r φk + a N 3/ k ( rk c ) ep N k k (37) where the radius function r k is given by r k ( k) + ( y yk) while ( k y k ) k n are called source points. The constants a and c are shape parameters. In the below we take the first set of φ k as trial functions which is known as a multi-quadric R [9 3] with N. In this eample we apply the multi-quadric radial basis function to solve the following non-linear PDE: 3 u u ( + y + a ) (38) where a was fied. The domain is an irregular domain with ρ( θ) (sin θ) ep(sin θ) + (cos θ) ep(cos θ). (39) The analytic solution is given by uy ( ) + y a which is singular on the circle with a radius a. ()

6 6 Journal of Marine Science and Technology Vol. No. (). E Residual error and stepsize E- E- Residual error by solving Δt Δt Residual error with a fied stepsize Error of u ig.. or eample : showing the convergence speed of residual errors and displaying the numerical error. Inserting Eq. (36) into Eq. (38) and placing some field points inside the domain to satisfy the governing equation and some points on the boundary to satisfy the boundary condition we can derive n NAEs to determine the n coefficients a k. The source points ( k y k ) k n are uniformly distributed on a contour given by R + ρ(θ k ) where θ k kπ/n. Under the following parameters R.5 c.5 ν 5 m. and t. in ig. we show the residual errors up to steps. It can be seen that in the residual-error curve there is a little oscillation and it decays very fast at the first few steps. The absolute error of numerical solution is plotted in ig. which is accurate with the maimum error being Now we test the effect by adjusting the stepsize. Under the following parameters R.5 c. ν 5 and m. in ig. 3 we compare the residual errors obtained by fiing a stepsize with t. which ehibits many irregular bursts as shown by the dashed line and that obtained by adjusting the stepsize as shown by a solid line which is smooth and decreased. The adaptive stepsize is also shown by the dasheddotted line up to 5 steps. - - y E Number of steps ig. 3. or eample comparing the residual errors obtained by a fied stepsize and an adaptive stepsize. 3. Eample 3 In this eample we apply the MECA to solve the following boundary value problem of non-linear elliptic equation [ ]: 3 uy ( ) + ω uy ( ) + εu( y ) py ( ). () While the eact solution is uy ( ) ( + y) + 3( y+ y) () 6 the eact p can be obtained by inserting the above u into Eq. (). y introducing a finite difference discretization of u at the grid points we can obtain i j ( u i+ j ui j+ ui j) + ( u i j+ ui j+ ui j ) ( ) ( y) + ω u + εu p. 3 i j i j i j (3) The boundary conditions can be obtained from the eact solution in Eq. (). Under the following parameters n n 3 t.5 ν 5 ω and ε. we compute the solution of the above system and compare them with the eact solution. In ig. we show the residual errors up to steps. The absolute errors of numerical solution are plotted in ig. of which the maimum error is smaller than.66 3.

7 C.-S. Liu: A Manifold-ased Algorithm 7.3 Stepsize Error of u. 6 8 ig. 5. or eample : displaying time stepsize and showing the convergence speed of residual errors. However the stepsize adjusting technique is computational time consuming ig.. or eample 3: showing the convergence speed of residual errors and displaying the numerical error..8 V. LIMITATION efore Eq. () we have mentioned that the governing equation of is not unique. Indeed it is not a necessary condition but a sufficient condition to satisfy Eq. (). Thus the following algorithm:.. ( t+ t) () t q() t t.6 y.8 () based-on Eq. (3) has some limitations. Obviously from the residual error curves as shown in igs. 3 and there is a common pattern that the curves decrease very fast at the beginning and then they all tend to be flattened with a very slow convergence without having an eponential convergence no longer. Another feature as obviously shown in igs. and is that the curves ehibit irregular bursts. To solve the latter problem we can employ the technology in Section III. by adjusting the time stepsize. As shown in ig. 5 we plot the adjustive time stepsize and residual error for eample. In the curve of residual error the bursts disappear. However it still has the first problem. More difficultly the time stepsize adjusting technique spends a lot of computational time to find t. In order to find the reason for a slow convergence of the residual error curve in its later stage we give the following analysis. Inserting Eqs. () and (5) into Eq. (3) we can derive where Qt () Qt ( + t) ( ) ( ) + a q t q t A a : in view of Eq. (6). rom Eq. () and the approimation of we have Q () t t Q( t+ t) Q() t (5) (6) qt () t [ Rt () ] (7)

8 8 Journal of Marine Science and Technology Vol. No. () a R (c) When R(t) tends to i.e. a tends to the dynamical force on the right-hand side is lost for the above equation. In ig. 6 we show a R and the residual error with respect to the number of steps. It can be seen that the above dynamic equation loses its pushing force and remains a large residual error due to a tending to and thus R tending to. VI. CONCLUSIONS A manifold-based eponentially convergent algorithm (MECA) was established in this paper. When we apply it to solve some D non-linear PDEs the vector-form was derived. Several numerical eamples of non-linear PDEs showed the efficiency and accuracy of MECA. The irregular bursts appeared in the residual error curve can be overcome by solving a scalar equation to find the adjustable fictitious time stepsize. However the governing Eq. (3) has faced a bottle-neck by only considering with the gradient vector as a unique source of dynamic force. An open problem is that how to construct a really eponentially convergent algorithm according to the setting of a manifold defined in Eq. (8). 6 8 ig. 6. or eample : displaying a showing R and (c) the convergence speed of residual errors. where the ratio Rt () is defined by Qt ( + t) Rt (). (8) Qt () As a requirement of Qt () > we need R(t) >. Thus through some manipulations Eq. (5) becomes ar() t (a + ) R() t + ( a + 8) Rt (). (9) 3 It is interesting that the above equation can be written as [ Rt ( ) ] [ art ( ) ]. (5) ecause R is a double roots and it is not the desired one we take Rt (). a A y using Eq. (7) Eq. () can now be written as ( t+ t) () t [ R() t ]. (5) (5) ACKNOWLEDGMENTS Taiwan s National Science Council project NSC-99-- E--7-MY3 granted to the author is highly appreciated. REERENCES. Allgower E. L. and Georg K. Numerical Continuation Methods: an Introduction Springer New York (99).. Atluri S. N. Methods of Computer Modeling in Engineering and Sciences Tech Science Press (). 3. Atluri S. N. Liu H. T. and Han Z. D. Meshless local Petrov-Galerkin (MLPG) mied collocation method for elasticity problems CMES: Computer Modeling in Engineering & Sciences Vol. pp. -5 (6).. Atluri S. N. Liu C.-S. and Kuo C. L. A modified Newton method for solving non-linear algebraic equations Journal of Marine Science and Technology Vol. 7 pp (9). 5. Atluri S. N. and Shen S. The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite and boundary element methods CMES: Computer Modeling in Engineering & Sciences Vol. 3 pp. -5 (). 6. Atluri S. N. and Zhu T. L. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics Computational Mechanics Vol. pp. 7-7 (998). 7. Atluri S. N. and Zhu T. L. A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation Computational Modeling and Simulation in Engineering Vol. 3 pp (998). 8. Chang C. W. and Liu C.-S. A fictitious time integration method for backward advection-dispersion equation CMES: Computer Modeling in Engineering & Sciences Vol. 5 pp (9). 9. Cheng A. H. D. Golberg M. A. Kansa E. J. and Zammito G. Eponential convergence and H-c multiquadric collocation method for partial differential equations Numerical Methods for Partial Differential Equations Vol. 9 pp (3).. Chi C. C. Yeih W. and Liu C.-S. A novel method for solving the Cauchy problem of Laplace equation using the fictitious time integration

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