We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for second order differential equations. An earlier result for single-parameter exponential fitting is re-examined and extended to multi-parameter, multi-order exponential fitting

Hollevoet, Davy

Van Daele, Marnix

Vanden Berghe, Guido

English

Ghent University Academic Bibliography

Multi-parameter exponentially fitted, P-stable ObrechkoffmethodsD. Hollevoet, M. Van Daele and G. Vanden BergheVakgroep Toegepaste Wiskunde en Informatica, Ghent University,Krijgslaan 281-S9, B-9000 Gent, BelgiumAbstract. We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for secondorder differential equations. An earlier result for single-parameter exponential fitting is re-examined and extended to multi-parameter, multi-order exponential fitting.Keywords: P-stability, Exponential fitting, Obrechkoff, Rational approximation, Multi-parameterPACS: 02.60.LjINTRODUCTIONIn [1], the authors considered symmetric two-step Obrechkoff methods of the formyn+1 2yn+ yn 1 =måj=1h2 jhb j0y(2 j)n+1+2b j1y(2 j)n +b j0y(2 j)n 1i(1)and showed that P-stability, a desirable property when dealing with periodic problems, can be guaranteed by imposingthat the stability function Rmm(v2) of the Obrechkoff method equals Re Pˆmm (iv), the real part of the [m=m]-Padéapproximation of eiv.Extending this result in [2], the technique of exponential fitting [3] is brought into the picture. Whereas a classicalObrechkoff method of order 2p has a polynomial fitting spacef1; t; t2; : : : ; t2p+1g(i.e. the method is exact for solutions that fall within that space), an exponentially fitted variant can have a so calledmixed-type fitting space (for any K > 0 and P< p such that K+2P= 2p 1)f1; t; t2; : : : ; tK ;em0t ; tem0t ; : : : ; tPem0tg (2)that contains one free parameter m0. If im0 2 R, this technique is also called trigonometric fitting because the fittingspace then contains pairs of sine and cosine. A more general approach often called multi-parameter exponential fitting,uses fitting spaces of the formf1; t; t2; : : : ; tK ;em0t ;em1t ; : : : ;emPtg: (3)and it is known that (2) is recovered when each m j approaches m0. In turn, the polynomial fitting space is found whenm0 ! 0. For this family of methods, basis functions y(t) = 1 and y(t) = t are always included in the fitting space dueto consistency and symmetry.It was shown that P-stability can also be guaranteed for exponentially fitted Obrechkoff methods of type (2) withim 2 R. This can be achieved by imposing that the stability function Rmm(v2) is the real part of an EF [m=m]-Padéapproximation Pˆmm (v) of eiv. This approximant is of the formPˆmm (v) =Vm(v)Vm( v) ; Vm(v) = 1+måj=1a j(m)v jand can be constructed by imposing8>>><>>>:¶ q¶xqF(x; t)j(x;t)=(0;m) = 0 q= 1 : : :K (4a)Re¶ q¶ tqF(x; t)j(x;t)=(i;m)= 0 q= 0 : : :P (4b)onF(x; t) := etxVm( v) Vm(v):In this paper, we want to show how this result can be simplified and how it extends to multi-parameter exponentialfitting.DERIVATIONClassical methodsThe linear stability theory for this type of methods (i.e. for second order differential equations) was introduced in[4] and is related to the test equationy00 = k2y; k 2 Rwith general solution y(t) = c1 cos(kt)+ c2 sin(kt). With this test equation, one obtains from (1)yn+1 2Rmm(v2)yn+ yn 1 = 0 (5)with v := kh and Rmm(v2) a rational function of order m=m. Idealiter, (5) is satisfied exactly for all solutions of the testequation, which would require Rmm(v2) cos(v).It seems obvious to choose the coefficients of the method such that Rmm(v2) is the [m=m]-Padé approximation ofcos(v). While this leads to methods with maximal order 4m, it does not provide P-stability. To obtain that, as shownin [1], the approximation cos(v) Re Pˆmm (iv) is better suited due to its boundedness on the entire real axis. It can beshown thatRe Pˆmm (iv)=12Pˆmm (iv)+ Pˆmm ( iv); (6)in which one recognizes the exponential form of cos(v). The property that Rmm(v2) is a rational function in v2 allowsto decompose it asRmm(v2) =12[Smm(iv)+Smm( iv)] ; (7)with Smm(v) also a rational function of degree m=m. Combining this with (6) leads to the requirement that Smm(iv) =Pˆmm (iv). From the properties of Padé approximations, the rules to impose on the stability function becomeS( j)mm(0) = 1; j = 0 : : :2m:This set of conditions is equivalent to (4) in case K = 2m+1.Exponentially fitted methodsSuppose we want to construct an Obrechkoff method with a fully-tuned multi-parameter fitting spacefem1t ;em2t ; : : : ;em2ptg;assuming distinct values for all m j. This can only be accomplished by imposing that (5) is fulfilled for v = iw j,w j := im jh, j = 1 : : :2p. The stability function still has the same structure, so we can decompose Rmm(v) as (7) againand obtain (Smm(w j) = ew j ; j = 1 : : :2p (8a)Smm( w j) = e w j ; j = 1 : : :2p: (8b)This time, there is no obvious symmetry, so both terms contribute 2p conditions. Since Smm is an expression with only2m unknowns, p can be at most bm=2c if no conditions coincide. When all these conditions are satisfied, Rmm(v) is arational interpolant of cos(v) through v= fv1; : : : ;v2pg.In [5], the same conditions are applied directly to Rmm(v2) and cos(v). This approach, however, does not necessarilyproduce P-stable methods.Symmetric exponential fittingSince we are considering a family of methods for second order differential equations, it is more sensible to talkabout trigonometric fitting with symmetric fitting spaces of type (2) or (3), instead of asymmetrical exponential fittingas above. It is known that the latter approach can be transformed into trigonometric fitting by takingmp+ j = m j; j = 1 : : : p:This step folds the two parts of (8) halfway onto each other, reducing the number of conditions to 2p and thusallowing m pairs of exponentials in the fitting space. Under these conditions, the interpolation difference functionE(w) := Smm(w)  ew has p pairs of roots, each symmetrical across the origin.Mixed-type, single-parameter EFThe conditions to impose on the stability function of a method with fitting space of the form (2) withK+2P= 2p 1can be found by taking an additional stepmr ! 0; j = 0 : : :Km j ! m0; j = K+1 : : :K+1+P:This causes the roots of E(w) to converge at w= 0 and w=hm0, leading to8>>><>>>:S( j)mm(0) = 1; j = 0 : : :KS( j)mm(hm0) = ehm0 ; j = 0 : : :PS( j)mm( hm0) = e hm0 ; j = 0 : : :P:Under these conditions, Rmm(v) becomes multi-point Padé approximation to cos(v) of order K in v= 0 and of order Pin v= ihm and v= ihm . It can be shown that when m0 is real, these condition coincide with (4).For example, the stability function of an Obrechkoff method with m = 2 and fitting space f1; t; t2; t3;emtg can bebuilt withS22(v) = ((m 2)em +2+m)v2+   m2+ emm2v+2emm2 2m2(( m+2)em  m 2)v2+( m2+ emm2)v 2emm2+2m2 : (9)The resulting Rmm(v) agrees with cos(v) on value, first and second derivative in v= 0 and on value only in v= im andv= im .Mixed-type, mixed-order, multi-parameter EFWhen m is large enough, any hybrid form is possible. It is possible to construct a method with a stability functionthat is an approximation to cos(v) in v = vk of order nk, k = 1 : : :r if åk nk  m. The conditions to impose on thecos v R33v2, m = 2, w = 9v2 4 6 8 10K1.0K0.500.51.0cos v R33v2, m = 2, w = 7v2 4 6 8 10K1.0K0.500.51.0FIGURE 1. Plots of R33(v;m = 2;w = 9) (left) and R33(v;m = 2;w = 7) (right). The stability function is a first order approxima-tion to cos(v) in v= 0, v= 2 and of zeroth order in v= 9 and v= 7 respectively.stability function are 8<: S( j)mm(wk) = ewk ; j = 0 : : :n jS( j)mm( wk) = e wk ; j = 0 : : :n j:with wk = ivk, for all vk under consideration.As an example, we have constructed an Obrechkoff method with fitting space f1; t;emt ; temt ;ewtg. From theabove statement, this requires that m = 3. Figure 1 shows two plots: on the left R33(v;m = 2;w = 9) and on theright R33(v;m = 2;w = 7), both along the real axis. The plots show that the stability function has the same value andderivative as cos(v) in v= 0 and v= 2, while there is only an agreement on value in v= 9 (left) and v= 7 (right).CONCLUSIONBased on the results for single-parameter exponential fitting, we derived the necessary conditions to construct thestability function of P-stable Obrechkoff methods of arbitrary order with multi-parameter, multi-order fitting spaces.The resulting rational functions can be interpreted as interpolations of cos(v) through different points fv1; : : : ;vmgwithmultiplicities fn1; : : : ;nmg.REFERENCES1. M. Van Daele, and G. Vanden Berghe, P-stable obrechkoff methods of arbitrary order for second-order differential equations,Numerical Algorithms 44, 115–131 (2007), ISSN 1017-1398.2. M. Van Daele, and G. Vanden Berghe, P-stable exponentially-fitted obrechkoff methods of arbitrary order for second-orderdifferential equations, Numerical Algorithms 46, 333–350 (2007).3. G. Ixaru, and G. Vanden Berghe, Exponential fitting, Kluwer Academic Publishers, Dordrecht, 2004.4. J. D. Lambert, and I. A. Watson, Symmetric multistip methods for periodic initial value problems, IMA Journal of AppliedMathematics 18, 189–202 (1976).5. J. P. Coleman, and L. G. Ixaru, P-stability and exponential-fitting methods for y” = f(x, y), IMA Journal of Numerical Analysis16, 179–199 (1996).

Multi-parameter exponentially fitted, P-stable Obrechkoff methods

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Multi-parameter exponentially fitted, P-stable Obrechkoff methods

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