In this paper we consider the approximate computation of isospectral flows based on finite integration methods( FIM) with radial basis functions( RBF) interpolation,a new algorithm is developed. Our method ensures the symmetry of the solutions. Numerical experiments demonstrate that the solutions have higher accuracy by our algorithm than by the second order Runge- Kutta( RK2) method

Li Kan

Liu Zhongyun

Zhang, Yulin

Universidade do Minho: RepositoriUM

第 35 卷 第 1 期2015 年 3 月数学理论与应用MATHEMATICAL THEOＲY AND APPLICATIONSVol． 35 No． 1Mar． 2015Finite Integration Methods for Isospectral FlowsLi Kan1 Liu Zhongyun1 Zhang Yulin2(1． School of Mathematics and Computer Science，Changsha University of Science and Technology，Changsha 410004，China)(2． Centro de Matemática，Universidade do Minho，Braga 4710 － 057，Portugal)Abstract In this paper we consider the approximate computation of isospectral flows based on finite integration meth-ods (FIM)with radial basis functions (ＲBF)interpolation，a new algorithm is developed． Our method ensures thesymmetry of the solutions． Numerical experiments demonstrate that the solutions have higher accuracy by our algo-rithm than by the second order Ｒunge － Kutta(ＲK2)method．Key words Isospectral flows Hermitian polynomial interpolation Finite integration method Ｒadial basis func-tion Ｒunge － Kutta method等谱流的有限积分法＊李 侃1 刘仲云1 张育林2(1．长沙理工大学数学与计算科学学院，长沙，410004)(2． Centro de Matemática，Universidade do Minho，Braga 4710 － 057，Portugal)摘 要 本文我们考虑基于径向基函数插值的有限积分法来近似求解等谱流问题，并提出了一种新的算法．我们的方法很好地保证了该问题解的对称性．此外，数值实验也证明了我们的新算法比二阶龙格库塔更精确．关键词 等谱流 Hermite多项式插值 有限积分法 径向基函数 Ｒunge － Kutta法1 IntroductionIsospectral flows can be characterized in terms of the matrix differential equationX' = ［B，X］，X(0)= X0， (1)where X，B∈ Ｒn×n and X0 is a given n × n initial matrix and X0 = XT0 ． B≡ B(t，X)is a matrix func-tion，which is allowed to depend on X and on the time t． The square brackets denote the commutatoror Lie bracket on matrices，namely［B，X］ = BX － XB，see for example［1］．＊Supported by the National Natural Science Foundation of China under Grant No． 11371075收稿日期:2015 年 1 月 20 日等谱流的有限积分法Numerical methods for solving (1)have intrigued the researchers for decades，see for example［1，2，3，6］and references therein．We are particularly interested in the case when B is skew － symmetric which frequently appearsin the typical isospectral flows，and will assume，unless explicitly mentioned，that BT = － B through-out the rest of the present paper．In this paper we focus on the direct solution to (1)by exploiting the finite integration methods．After briefly reviewing the finite integration method with radial basis function in next section，see forexample［7］，we develop a new algorithms based on FIM for solving (1)in section 3． Then somenumerical examples are presented in section 4，in which the numerical results demonstrate that ouralgorithm yield a much higher degree of accuracy than ＲK2 method．2 Finite integration methodIn this section we review the FIM with radial basis functions (ＲBF) ，which can be outlined asfollows． The field variable f(t)in the interval［a，b］can be interpolated over a number of randomlydistributed nodes{tk}Nk = 0 and nodal values{fk}Nk = 0 with t0 = a and tN = b，asf^(t)= ∑Ni = 0Ｒi(t，ti)αi +∑Qq = 0Pq(t)βq， with f^(tk)= fk，k = 0，1，…，N， (2)where {Ｒi(t，ti) }Ni = 0 is a set of radial basis functions centred at t，{Pq(t) }Qq = 0 is a set of the polyno-mial basis，{αi}Ni = 0 and{βq}Qq = 0 are the coefficients of Ｒ(t)and mP(t)respectively． The polynomialterm has to satisfy an extra requirement that guarantees unique approximation of a function as follows∑Ni = 1Pq(ti)αi = 0，q = 0，1，2，…，Q．So we can get the coefficienta = Ｒ－10 ［I － P(PTＲ－10 P)－1PTＲ－10 ］f，b = (PTＲ－10 P)－1PTＲ－10 f， (3)where a = ［α0，α1，…，αN］T，b = ［β0，β1，…，βN］T，f = ［^f0，^f1，…，^fN］T，I denotes the identitymatrix，Ｒ0 = (Ｒi(tk，ti) ) ，P = (Pq(ti) ) (q = 1，2，…，Q and k，i = 1，2，…，N)． Substituting a，b in (3)into (2)yieldsf^(t)= Ｒ(t)Ｒ－10 ［I － P(PTＲ－10 P)－1PTＲ－10 ］f + P(t) (PTＲ－10 P)－1PTＲ－10 f = ∑Ni = 1i(t)^f i，(4)where i(t) ，i = 0，1，…，N，are called shape functions which have the Kronecker － delta property，15数学理论与应用Consequently we obtain its first order integration easily asF(t)= ∫taf^(t)dt = ∑Ni = 1^i(t)^f i， (5)where^i(t)= ∫tai(t)dt． (6)Let Fk = ∫tkaf^(t)dt，F =［F0，F1，…，FN］T，and A = (^ki) ，with ^ik = ^i(tk) ，then the dis-crete finite integration of the first order can be rewritten asF = Af． (7)3 Finite integration method for isospectral flowsIn this section，we consider the computation of isospectral flows (1)with FIM．Assume that Xm is obtained． We first introduce a uniform division to［tm，tm+1］for ν subinter-vals，that is to say that t(k)m = t0 + δk，δ = (tm+1 － tm)/ν，k = 0，1，2，…，ν are collocation pointsin the region of t∈［tm，tm+1］，then a direct integration shows thatX(1)m+1 － Xm = ∫tm+δtm(BmXm － XmBm)dt，X(2)m+1 － Xm = ∫tm+2δtm(B(1)m+1X(1)m+1 － X(1)m+1B(1)m+1)dt，X(ν)m+1 － Xm = ∫tm+1tm(B(ν－1)m+1 X(ν－1)m+1 － X(ν－1)m+1 B(ν－1)m+1 ) dt．(8)Thus we obtain ν nodal approximative values {X(k)m+1}νk = 1 and their corresponding derivatives{X'(k)m+1}νk = 1 which are defined byX'(k)m+1 = B(k)m+1X(k)m+1 － X(k)m+1B(k)m+1 ． (9)Then letX^m+1(t)= Hermite (X(1)m+1，X(2)m+1，…，X(ν)m+1) (10)be the piecewise hermitian interpolation polynomial of Xm+1(t)of degree 2ν － 1 at(tm，tm+1)． Thuswe just get an approximate function of integral function f(X，t)= BX － XB．In order to obtain more accurate approximation Xm+1 of X(tm+1)than X^m+1，we next equally par-25等谱流的有限积分法tition the interval［tm，tm+1］into N subintervals． Let t(k)m = tm + Δk，Δ = (tm+1 － tm)/N，i = 0，1，2，…，N，are collocation points in the region of t ∈［tm，tm+1］． After we have N + 1 nodal valuesX^(0)m+1，X^(1)m+1，…，X^(N)m+1，then by (5) ，we obtain a sequence of matrices{X(k)m+1}Nk = 0 ．Finally，we denote X(N)m+1 = Xm+1 ． Thus we have got a sequence of approximate matrices ofX(tm+1) ，for m = 0，1，…，which are the approximate solutions of (1)at t = 1，…．We conclude the above process in the following algorithm．ALGOＲITHM［FIM with ＲBF］1． Input X0，t0 ＜ t1 ＜ … ＜ tm ＜ tm+1 ＜ …，for m = 0，1，…，2． partition［tm，tm+1］into νm subintervals，3． for k = 1，2，…，νm，compute{X(k)m+1}νmk = 1 by (8)and X'(k)m+1 by (9) ，4． construct hermitian interpolation polynomial of X^m+1(t)of degree 2νm － 1 at (tm，tm+1)by(10) ，5． randomly partition［tm，tm+1］into Nm subintervals: ［tm，t(1)m ］，…，［t(Nm－1)m ，tm+1］，6． input radial basis functions{Ｒi(t，ti) }Nmi = 0 and polynomial basis{Pq(t) }Qmq = 0，7． for k = 0，1，…，Nm，compute f(k)m+1 = X^m+1(t(k)m ) ，where t(0)m = tm and t(Nm)m = tm+1，and formfm+1 = ［f(0)m+1，…，f(k)m+1，…，f(Nm)m+1 ］T，8． compute a = ［α0，α1，…，αN］T，b = ［β0，β1，…，βN］T by (3) ，9． form f^(m+1)(t)in (2)and shape functions (m+1)i in(4) ，10． compute ^(m+1)ik = ^(m+1)i (t(k)m )defined in (6)and Am+1 = (^(m+1)ik )Nmi，k = 0，11． compute Fm+1 = ［F(0)m+1，…，F(k)m+1，…，F(Nm)m+1 ］T by (7) ，12． set F(Nm)m+1 = Xm+1 (This is the approximation of the solution of (1)at t = tm+1)．4 Examples and numerical resultsIn our tests，the matrix B has the formB = X(t)－ － X(t)+ ．Moreover，we consider the case of uniformly distributed interpolation collocation points and νm= 2，as well as equally distributed approximative nodes and Nm = 10． And we take into account thatＲi(t，ti)= c2 + (t － ti)槡 2，Pq(t)= tq and the free parameter c = 1 /Nm，Q = 7． Additionally，thetypical finite difference method(ＲK2)is compared to demonstrate the accuracy of the finite integra-35数学理论与应用tion method． We compare the error in relative similarity for three strategies，presentingei =vi+1 － vi ∞vi ∞，where vi is a vector that is composed of eigenvalues sorted accordingly to increasing size of the ap-proximation Xi，for i = 0，1，…．Example 4． 1 We have used the same 10 × 10 randomly generated symmetric matrix X0 forstepsize Δ = 1 /10． The Table 4． 1 denotes the numerical results obtained for ＲK2 method and FIM(ＲBF)respectively．Table 1 Stepsize Δ = 1 /5，relative similarity errors (Er)for different methodsei ＲK2 ＲBFe1 1． 0951e － 001 4． 0797e － 002e2 1． 6081e － 001 7． 4533e － 002e3 4． 0796e － 002 1． 9994e － 003e4 1． 6289e － 002 2． 3372e － 003e5 5． 4959e － 003 4． 7456e － 004e6 1． 7184e － 003 2． 6977e － 004e7 5． 1724e － 004 1． 4034e － 004e8 2． 4323e － 004 6． 0153e － 005e9 1． 4415e － 004 4． 4094e － 005e10 8． 7037e － 005 3． 3145e － 005Example 4． 2 We exploit the same way to obtain X0 different from that in example 4． 1 forstepsize Δ = 1 /5． The numerical results as follows:Table 1 Stepsize Δ = 1 /5，relative similarity errors (Er)for different methodsei ＲK2 ＲBFe1 1． 8844e － 001 4． 2309e － 002e2 6． 7262e － 002 7． 3751e － 002e3 4． 0796e － 002 1． 6579e － 003e4 4． 6225e － 002 2． 4193e － 003e5 3． 2873e － 002 4． 6971e － 004e6 2． 4049e － 002 6． 8787e － 005e7 1． 8045e － 002 3． 1867e － 005e8 1． 4104e － 002 1． 6507e － 005e9 1． 1694e － 002 9． 6746e － 006e10 1． 0492e － 002 5． 8707e － 00645等谱流的有限积分法It is apparent that we can get more ideal results with FIM than FDM． And in our tests，thereare some problems deserve to be mentioned． When Δ is smaller(for instance Δ = 0． 01) ，the solu-tion was presented with NAN using ＲBF and the result with ＲK2 method is better．Ｒeference［1］M． E Calvo，A． Iserles，and A． Zanna，Ｒunge － Kutta methods for orthogonal and isospectral flows，AppliedNumerical Mathematics，1996，22:153 － 163．［2］M． E． Calvo，A． Iserles and A． Zanna，Numerical solution of isospectral flows，Mathematics of Computation，1997，Vol． 66:1461 － 1486．［3］M． E Calvo，A． Iserles and A． Zanna，Semi － explicit methods for isospectral flows，Advances in Computation-al Mathematics，2001，14:1 － 24．［4］G． H． Golub，C． F． Van Loan，Matrix Computations，third ed．，Johns Hopkins University Press，Baltimore，London，1996．［5］M． Li，Y． C． Hon，T． Korakianitis，and P． H． Wen，Finite integration method for nonlocal elastic bar understatic and dynamic loads，Engineering Analysis with Boundary Elements，2013，37:842 － 849．［6］D． V． Watkins，Isospectral flows SIAM Ｒeview，1984，Vol． 26:379 － 391．［7］P． H． Wen，Y． C． Hon M． Li，and T． Korakianitis，Finite integration method for partial differential equa-tions，Applied Mathematical Modelling，2013，37:10092 － 10106．55

Finite integration methods for isospectral flows

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Finite integration methods for isospectral flows

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