The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced by R. Weiner et al [6, 7, 11, 12, 13] for solving different types of problems either in sequential or parallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces with dimension six. Finally, some numerical experiments are presented to show the behaviour of the new peer schemes for some periodic problems

Calvo, M

Montijano, JI

Rández, L

Van Daele, Marnix

English

M. Calvo

J. I. Montijano

L. Rández

M. Van Daele

Crossref

Exponentially fitted fifth-order two-step peer explicit methods

Calvo, M.

Montijano, J. I.

Rández, L.

Van Daele, M.

Repositorio Universidad de Zaragoza

Exponentially Fitted Fifth-Order Two-Step Peer ExplicitMethodsM. Calvo∗, J. I. Montijano∗, L. Rández∗ and M. Van Daele†∗Instituto Universitario de Matemáticas y Aplicaciones.Departamento de Matemática Aplicada.Facultad de Ciencias. Universidad de Zaragoza.Pedro Cerbuna 12, 50009 Zaragoza†Vakgroep Toegepaste Wiskunde en Informatica.Universiteit Gent, Krijgslaan 281-59.B9000 Gent, Belgium.Abstract. The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differentialsystems were introduced by R. Weiner et al [6, 7, 11, 12, 13] for solving different types of problems either in sequential orparallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaceswith dimension six. Finally, some numerical experiments are presented to show the behaviour of the new peer schemes forsome periodic problems.Keywords: Exponential fitting, Peer methods, oscillatory initial value problemsPACS: 02.60.LjINTRODUCTIONWe consider the numerical solution of IVPs for first order differential systemsddty(t) = f (t,y(t)), t ∈ [t0, t0 +T ], y(t0) = y0 ∈ Rm, (1)where for simplicity f (t,y) is assumed to be sufficiently smooth, so that the IVP (1) has a unique solution. Furthermore,some knowledge of the behaviour of their unique global solution is known in advance. In the case that the solution of(1) has an oscillatory behaviour and further we know an estimate of the frequency, some modified Runge-Kutta (RK)methods using this information, usually called trigonometrically fitted or more generally exponentially fitted methods[3, 2, 4, 5, 9, 10] have been proposed in the last years to improve their accuracy and efficiency over standard RKmethods.For explicit RK methods the stage order is limited to one and this implies serious restrictions in the dimensionalityof the fitting space. On the other hand, linear multistep methods do not have such a limitation, as shown for examplein the early paper of Gautschi [3]. In this case, with k steps, a method can be fitted to k+1 dimensional spaces.Here, we consider explicit two-step peer methods introduced by R. Weiner, et al [6, 7] as an alternative to classicalRunge–Kutta (RK) and multistep methods attempting to combine the advantages of these two classes of methods.In particular fitted three-stage peer methods based on the previous schemes given in [1] to some fitting spaces areconstructed.Some numerical experiments are presented to show the performance of the above fitted methods for problems withoscillatory solutions. The proposed methods are compared to exponentially fitted Adams-Bashforth-Moulton methodswith the same order.FITTED TWO STEP PEER METHODSIt will be sufficient to consider in (1) scalar equations (m = 1) and then a s-stage peer scheme can be expressed invector form asY1 = A Y0 +h B F0 +h R F1, (2)Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014)AIP Conf. Proc. 1648, 150015-1–150015-4; doi: 10.1063/1.4912445© 2015 AIP Publishing LLC 978-0-7354-1287-3/$30.00150015-1where the s-dim vector Y j = {Yj,l} includes approximations to the solution of (1) at t j + clh, l = 1, . . . ,s andYk =(Yk,1, . . . ,Yk,s)T ∈ Rs,e = (1, . . . ,1)T ∈ Rs,(3)c = (c1, . . . ,cs)T ∈ Rs,Fk = f (tke+hc,Yk) =(f (tk +hc j,Yk, j))sj=1 ∈ Rs,where A ∈ Rs×s and B ∈ Rs×s full matrices and R ∈ Rs×s strictly lower triangular are the free parameters that definethe method and the set of admissible fixed nodes c j, j = 1, . . . ,s satisfies that |ci − c j| = 0,1 for all i = j.It is worth to note that (2) is a multivalued method and 0-stability requires that the matrix A satisfies the root’scondition. In this work we consider only methods with the requirement that the eigenvalues λ j(A) of A,λ1(A) = 1, λ j(A) = 0, j = 2, . . . ,s. (4)Furthermore, following the ideas of [1], in order to simplify the derivation of the fitting methods and to get methodswith high stage order, we will take A with the formA = P−1 Â P, (5)with P= (pi j)∈Rs×s a lower triangular matrix with ones at the diagonal, and Â= (aˆi j)∈Rs×s upper triangular whosediagonal is diag(Â) = (1,0, . . . ,0), that clearly satisfy (4).To study the local discretization error, we associate to (2) the linear s-dimensional vector valued operator L [ϕ;h]defined on sufficiently smooth scalar real or complex functions ϕ byL [ϕ;h](t) = ϕ((t+h)e+hc)−A ϕ(te+hc)−h B ϕ˙(te+hc)−h R ϕ˙((t+h)e+hc). (6)For our class of strongly 0-stable peer methods where A is given by (5) we haveL [ϕ ;h](t0) = P−1 L̂ [ϕ ;h](t0), (7)withL̂ [ϕ;h](t0) = Z(t+h)− Â Z(t)−hB̂ Z˙(t)−hR̂ Z˙(t+h), (8)and Z(t) = P ϕ(te+hc). In view of (7) the method (2) is fitted to the functional space Fq at t0 with step size h iffL̂ [ϕ ;h](t0) = 0, ∀ϕ ∈Fq. (9)We can state the next important result:Theorem.- Suppose that for a given set of admissible fixed nodes and constant matrix P the standard two-step peermethod (2), (5) with s stages has a unique solution with stage order 2s−1, then:1. For any linear space F2s−1 = 〈1,ϕ1(t), . . . ,ϕ2s−1(t)〉 there exists a unique s-stage two step peer method fitted tothis space for h sufficiently small. This peer method has the same nodes and P-matrix as the standard method andthe coefficientsÂF = Â(t0,h), B̂F = B̂(t0,h), R̂F = R̂(t0,h),may depend (apart from the fitting space) on t0 and h.2. If F2s−1 is a separable basis the coefficients are independent of t0.THREE-STAGE EXPONENTIALLY FITTED PEER METHODSIn this section, we consider exponential fitted variants for the fifth order peer scheme deduced in [1], which are fittedto the particular spaces• F5 = 〈1, t, t2, t3, t4, t5〉150015-2• F5(ω) = 〈1, t, t2, t3,sin(ω t),cos(ω t)〉• F5(ω1,ω2) = 〈1, t,sin(ω1 t),cos(ω1 t),sin(ω2 t),cos(ω2 t)〉where ω,ω1,ω2 are fixed positive parameters.Due to the linear, explicit, multistep nature of each stage of the PEER schemes, we use predictor-corrector pairs oflinear multistep methods for comparison. We choose the classical four-step Adams-Bashforth-Moulton pair of order 5(denoted by AM5), the fitted versions of the previous scheme toF5(ω) (denoted by AM5(ω)), toF5(ω1,ω2) (denotedby AM5(ω1, ω2)). The fitted multistep methods have been obtained following the ideas in [3].We have implemented the AM5 schemes in the mode PECE, that requires only 2 function evaluations per step. Forthe PEER3 methods, 3 function evaluations per step are needed, so in order to be able to compare the computationalcost (total number of function evaluations), we have applied PEER3 methods with stepsizes 50 % larger than those ofthe AM5 methods.The criterion used in the numerical comparisons is the maximum global error in the solution over the wholeintegration interval. All computations were carried out in quadruple precision arithmetic (32 significant digits ofaccuracy) on a PC computer running PYTHON.ProblemWe consider the two-dimensional semi-linear oscillatory second-order IVPq′′(t) =−12(w2 + k2+1 w2− k2−1w2− k2−1 w2 + k2 +1)q(t)+k22(q1(t)−q2(t))3(1−1)with initial conditionsq(0) =( −1/2−1/2), q′(0) =12( −√2−w√2−w)and the parameters w> 0, 0≤ k< 1. This problem represents a simple model consisting of two point masses connectedby a soft nonlinear spring and a stiff linear one, and its exact periodic solution can be written in terms of the Jacobi’selliptic functions asq(t) =√22(cos(π/4+wt)− sn(t,k)cos(π/4+wt)+ sn(t,k))In our numerical experiments we have taken w = 2, k = 0.05, tend = 100, and h = 1/(5× 2i), i = 1, . . . ,5 and thenumerical results are presented in Figure 1.????????????????????????????????????????????????????????????????????????????????????????????????????????????????FIGURE 1. Efficiency plot of fifth-order schemes for the two-dimensional semi-linear oscillatory problem with ω1 = ω = 2,ω2 = 1, k = 0.05In this test problem, the efficiency of the peer schemes considered is slighty superior to the corresponding multistepmethods. Also, the methods fitted with two frequencies ω1 = 2, ω2 = 1 perform much better than the correspondingschemes fitted only to one frequency.150015-3REFERENCES1. M. Calvo, J.I. Montijano, L. Rández and M.Van Daele, On the derivation of explicit two step peer methods, Appl. Numer.Math. 61 (2011), n. 4, 395–409.2. J.M. Franco, Runge-Kutta methods adapted to the numerical integration of oscillatory problems, Appl. Numer. Math. 50(2004) 427–443.3. W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3,(1961) 381–397.4. L.Gr. Ixaru, and G. Vanden Berghe, Exponential Fitting, Mathematics and Its Applications vol. 568, Kluwer, Dordrecht, 2004.5. B. Paternoster, Runge–Kutta (-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials.Applied Numerical Matehmatics 28, 401–412 (1998).6. B.A. Schmitt, R. Weiner, Parallel two-step W-methods with peer variables, SIAM J. Numer. Anal. 42 (2004), no. 1, 265–282 .7. B.A. Schmitt, R. Weiner, K. Erdmann, Implicit parallel peer methods for stiff initial value problems, Appl. Numer. Math. 53(2005), no. 2-4, 457–470.8. B.A. Schmitt, R. Weiner, S. Jebens, Parameter optimization for explicit parallel peer two-step methods. Appl. Numer. Math.59, 769–782 (2009).9. G. Vanden Berghe, H. De Meyer, M. Van Daele and T. Van Hecke , Exponentially-fitted explicit Runge–Kutta methods.Comput. Phys. Comm. 123 7–15 (1999).10. G. Vanden Berghe and M. Van Daele , Trigonometric polynomial or exponential fitting approach?. Journal of Computationaland Applied Mathematics, 233, 4, 969–979, (2009).11. R. Weiner, B.A. Schmitt, H. Podhaisky, Linearly-implicit two-step methods and their implementation in Nordsieck form,Appl. Numer. Math. 56 (2006), no. 3–4, 374–387.12. R. Weiner, B.A. Schmitt, H. Podhaisky, S. Jebens, Superconvergent explicit two-step peer methods, Journal of Computationaland Applied Mathematics, 223, 753–764 (2009).13. R. Weiner, K. Biermann, B.A. Schmitt, H. Podhaisky, Explicit two-step peer methods, Computers and Mathematics withApplications 55, 609–619 (2008).150015-4AIP Conference Proceedings is copyrighted by AIP Publishing LLC (AIP). Reuse of AIPcontent is subject to the terms at: http://scitation.aip.org/termsconditions. For moreinformation, see http://publishing.aip.org/authors/rights-and-permissions.

Ghent University Academic Bibliography

Exponentially Fitted Fifth-Order Two-Step Peer ExplicitMethodsM. Calvo∗, J. I. Montijano∗, L. Rández∗ and M. Van Daele†∗Instituto Universitario de Matemáticas y Aplicaciones.Departamento de Matemática Aplicada.Facultad de Ciencias. Universidad de Zaragoza.Pedro Cerbuna 12, 50009 Zaragoza†Vakgroep Toegepaste Wiskunde en Informatica.Universiteit Gent, Krijgslaan 281-59.B9000 Gent, Belgium.Abstract.The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systemswere introduced by R. Weiner et al [6, 7, 11, 12, 13] for solving different types of problems either in sequential or parallelcomputers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces withdimension six. Finally, some numerical experiments are presented to show the behaviour of the new peer schemes for someperiodic problems.Keywords: Exponential fitting, Peer methods, oscillatory initial value problemsPACS: 02.60.LjINTRODUCTIONWe consider the numerical solution of IVPs for first order differential systemsddty(t) = f (t,y(t)), t ∈ [t0, t0+T ], y(t0) = y0 ∈ Rm, (1)where for simplicity f (t,y) is assumed to be sufficiently smooth, so that the IVP (1) has a unique solution. Furthermore,some knowledge of the behaviour of their unique global solution is known in advance. In the case that the solution of(1) has an oscillatory behaviour and further we know an estimate of the frequency, some modified Runge-Kutta (RK)methods using this information, usually called trigonometrically fitted or more generally exponentially fitted methods[3, 2, 4, 5, 9, 10] have been proposed in the last years to improve their accuracy and efficiency over standard RKmethods.For explicit RK methods the stage order is limited to one and this implies serious restrictions in the dimensionalityof the fitting space. On the other hand, linear multistep methods do not have such a limitation, as shown for examplein the early paper of Gautschi [3]. In this case, with k steps, a method can be fitted to k+1 dimensional spaces.Here, we consider explicit two-step peer methods introduced by R. Weiner, et al [6, 7] as an alternative to classicalRunge–Kutta (RK) and multistep methods attempting to combine the advantages of these two classes of methods.In particular fitted three-stage peer methods based on the previous schemes given in [1] to some fitting spaces areconstructed.Some numerical experiments are presented to show the performance of the above fitted methods for problems withoscillatory solutions. The proposed methods are compared to exponentially fitted Adams-Bashforth-Moulton methodswith the same order.FITTED TWO STEP PEER METHODSIt will be sufficient to consider in (1) scalar equations (m = 1) and then a s-stage peer scheme can be expressed invector form asY1 = A Y0+h B F0+h R F1, (2)where the s-dim vector Y j = {Yj,l} includes approximations to the solution of (1) at t j + clh, l = 1, . . . ,s andYk =(Yk,1, . . . ,Yk,s)T ∈ Rs,e = (1, . . . ,1)T ∈ Rs,(3)c = (c1, . . . ,cs)T ∈ Rs,Fk = f (tke+hc,Yk) =(f (tk +hc j,Yk, j))sj=1 ∈ Rs,where A ∈ Rs×s and B ∈ Rs×s full matrices and R ∈ Rs×s strictly lower triangular are the free parameters that definethe method and the set of admissible fixed nodes c j, j = 1, . . . ,s satisfies that |ci− c j| 6= 0,1 for all i 6= j.It is worth to note that (2) is a multivalued method and 0-stability requires that the matrix A satisfies the root’scondition. In this work we consider only methods with the requirement that the eigenvalues λ j(A) of A,λ1(A) = 1, λ j(A) = 0, j = 2, . . . ,s. (4)Furthermore, following the ideas of [1], in order to simplify the derivation of the fitting methods and to get methodswith high stage order, we will take A with the formA = P−1 Â P, (5)with P= (pi j)∈Rs×s a lower triangular matrix with ones at the diagonal, and Â= (aˆi j)∈Rs×s upper triangular whosediagonal is diag(Â) = (1,0, . . . ,0), that clearly satisfy (4).To study the local discretization error, we associate to (2) the linear s-dimensional vector valued operator L [ϕ;h]defined on sufficiently smooth scalar real or complex functions ϕ byL [ϕ;h](t) = ϕ((t+h)e+hc)−A ϕ(te+hc)−h B ϕ˙(te+hc)−h R ϕ˙((t+h)e+hc). (6)For our class of strongly 0-stable peer methods where A is given by (5) we haveL [ϕ;h](t0) = P−1 L̂ [ϕ;h](t0), (7)withL̂ [ϕ;h](t0) = Z(t+h)− Â Z(t)−hB̂ Z˙(t)−hR̂ Z˙(t+h), (8)and Z(t) = P ϕ(te+hc). In view of (7) the method (2) is fitted to the functional spaceFq at t0 with step size h iffL̂ [ϕ;h](t0) = 0, ∀ϕ ∈Fq. (9)We can state the next important result:Theorem.- Suppose that for a given set of admissible fixed nodes and constant matrix P the standard two-step peermethod (2), (5) with s stages has a unique solution with stage order 2s−1, then:1. For any linear spaceF2s−1 = 〈1,ϕ1(t), . . . ,ϕ2s−1(t)〉 there exists a unique s-stage two step peer method fitted tothis space for h sufficiently small. This peer method has the same nodes and P-matrix as the standard method andthe coefficientsÂF = Â(t0,h), B̂F = B̂(t0,h), R̂F = R̂(t0,h),may depend (apart from the fitting space) on t0 and h.2. IfF2s−1 is a separable basis the coefficients are independent of t0.THREE-STAGE EXPONENTIALLY FITTED PEER METHODSIn this section, we consider exponential fitted variants for the fifth order peer scheme deduced in [1], which are fittedto the particular spaces• F5 = 〈1, t, t2, t3, t4, t5〉• F5(ω) = 〈1, t, t2, t3,sin(ω t),cos(ω t)〉• F5(ω1,ω2) = 〈1, t,sin(ω1 t),cos(ω1 t),sin(ω2 t),cos(ω2 t)〉where ω,ω1,ω2 are fixed positive parameters.Due to the linear, explicit, multistep nature of each stage of the PEER schemes, we use predictor-corrector pairs oflinear multistep methods for comparison. We choose the classical four-step Adams-Bashforth-Moulton pair of order 5(denoted by AM5), the fitted versions of the previous scheme toF5(ω) (denoted by AM5(ω)), toF5(ω1,ω2) (denotedby AM5(ω1, ω2)). The fitted multistep methods have been obtained following the ideas in [3].We have implemented the AM5 schemes in the mode PECE, that requires only 2 function evaluations per step. Forthe PEER3 methods, 3 function evaluations per step are needed, so in order to be able to compare the computationalcost (total number of function evaluations), we have applied PEER3 methods with stepsizes 50 % larger than those ofthe AM5 methods.The criterion used in the numerical comparisons is the maximum global error in the solution over the wholeintegration interval. All computations were carried out in quadruple precision arithmetic (32 significant digits ofaccuracy) on a PC computer running PYTHON.ProblemWe consider the two-dimensional semi-linear oscillatory second-order IVPq′′(t) =−12(w2+ k2+1 w2− k2−1w2− k2−1 w2+ k2+1)q(t)+k22(q1(t)−q2(t))3(1−1)with initial conditionsq(0) =( −1/2−1/2), q′(0) =12( −√2−w√2−w)and the parameters w> 0, 0≤ k < 1. This problem represents a simple model consisting of two point masses connectedby a soft nonlinear spring and a stiff linear one, and its exact periodic solution can be written in terms of the Jacobi’selliptic functions asq(t) =√22(cos(pi/4+wt)− sn(t,k)cos(pi/4+wt)+ sn(t,k))In our numerical experiments we have taken w = 2, k = 0.05, tend = 100, and h = 1/(5× 2i), i = 1, . . . ,5 and thenumerical results are presented in Figure 1.In this test problem, the efficiency of the peer schemes considered is slighty superior to the corresponding multistepmethods. Also, the methods fitted with two frequencies ω1 = 2, ω2 = 1 perform much better than the correspondingschemes fitted only to one frequency.REFERENCES1. M. Calvo, J.I. Montijano, L. Rández and M.Van Daele, On the derivation of explicit two step peer methods, Appl. Numer.Math. 61 (2011), n. 4, 395–409.2. J.M. Franco, Runge-Kutta methods adapted to the numerical integration of oscillatory problems, Appl. Numer. Math. 50(2004) 427–443.3. W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3,(1961) 381–397.4. L.Gr. Ixaru, and G. Vanden Berghe, Exponential Fitting, Mathematics and Its Applications vol. 568, Kluwer, Dordrecht, 2004.5. B. Paternoster, Runge–Kutta (-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials.Applied Numerical Matehmatics 28, 401–412 (1998).6. B.A. Schmitt, R. Weiner, Parallel two-step W-methods with peer variables, SIAM J. Numer. Anal. 42 (2004), no. 1, 265–282 .7. B.A. Schmitt, R. Weiner, K. Erdmann, Implicit parallel peer methods for stiff initial value problems, Appl. Numer. Math. 53(2005), no. 2-4, 457–470.104−14−13−12−11−10−9−8−7−6−5−4−3nfcnglobal error  PEER3 standardPEER3(w1,w2)PEER3(w)AM5 standardAM5 (w1,w2)AM5 (w)FIGURE 1. Efficiency plot of fifth-order schemes for the two-dimensional semi-linear oscillatory problem with ω1 = ω = 2,ω2 = 1, k = 0.058. B.A. Schmitt, R. Weiner, S. Jebens, Parameter optimization for explicit parallel peer two-step methods. Appl. Numer. Math.59, 769–782 (2009).9. G. Vanden Berghe, H. De Meyer, M. Van Daele and T. Van Hecke , Exponentially-fitted explicit Runge–Kutta methods.Comput. Phys. Comm. 123 7–15 (1999).10. G. Vanden Berghe and M. Van Daele , Trigonometric polynomial or exponential fitting approach?. Journal of Computationaland Applied Mathematics, 233, 4, 969–979, (2009).11. R. Weiner, B.A. Schmitt, H. Podhaisky, Linearly-implicit two-step methods and their implementation in Nordsieck form,Appl. Numer. Math. 56 (2006), no. 3–4, 374–387.12. R. Weiner, B.A. Schmitt, H. Podhaisky, S. Jebens, Superconvergent explicit two-step peer methods, Journal of Computationaland Applied Mathematics, 223, 753–764 (2009).13. R. Weiner, K. Biermann, B.A. Schmitt, H. Podhaisky, Explicit two-step peer methods, Computers and Mathematics withApplications 55, 609–619 (2008).

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Exponentially fitted fifth-order two-step peer explicit methods

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