# Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Authors: Samuel N. Jator; Kindyl L. King
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.


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## Body

## 1. Introduction

General second-order differential equations frequently arise in several areas of science and engineering, such as celestial mechanics, quantum mechanics, control theory, circuit theory, astrophysics, and biological sciences. Several of these differential equations have oscillatory solutions, for instance, the initial value problem (IVP)(1)y′′+ℵy=gt,y,y′,yt0=y0,y′t0=y0′,t0≤t≤tN,where g:R×Rd×Rd→Rd,  N>0 is an integer, {y,y′}ϵRd, d is the dimension of the system, ℵ is a real nonsingular diagonalizable d×d constant matrix having large eigenvalues, and ℵ≫1 has been studied (see [1–9]). Special cases of (1) have also been extensively discussed in the literature ([10–14], Hairer [15]).The method proposed in this paper can solve (1), as well as the general second-order IVP(2)y′′=ft,y,y′,yt0=y0,y′t0=y0′,t0≤t≤tN,where f:R×Rd×Rd→Rd,  N>0 is an integer, and d is the dimension of the system. Equation (2) is conventionally solved by converting it into an equivalent first-order system of double dimension and then solved using standard methods that are available in the literature for solving systems of first-order IVPs (see Lambert [16], Hairer et al. [17], and Brugnano et al. [18]). In general, these methods are implemented in a step-by-step fashion in which, on the partition SN, an approximation yn is obtained at tn only after an approximation at tn-1 has been computed, where for some constant step-size h=tN-t0/N and integer N>0,(3)SN≔t0<t1<…<tN,tn=tn-1+h,n=1,…,N.Several methods have been proposed for directly solving the special second-order IVP, where the functionf does not depend on y′. It has been shown that such methods have the advantages of requiring less storage space and fewer number of function evaluations (see Hairer [15], Hairer et al. [19], Lambert et al. [20], and Twizell et al. [21]). Nevertheless, these methods are implemented as predictor-corrector methods and require starting values as well. In which case, the cost of implementation is increased, especially, for higher order methods.In this paper, we propose a high-order BHM of order 11 which provides greater accuracy even when larger step-sizes are used. We show that the BHM of order 11 is applied to directly solve Hamiltonian systems as well as the general form (2). The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. In this way, the method is self-starting and implemented without predictors (see Jator and Oladejo [22], Jator et al. [9], Jator [7], and Ngwane and Jator [6, 23, 24]). It is shown that the method can also be used to directly solve non-Hamiltonian systems with embedded first derivatives as well as partial differential equations via the method of lines. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.The article is organized as follows. In Section2, we derive our hybrid method and specify the coefficients as well. We discuss the properties and implementation of the method in Section 3. Numerical examples are given in Section 4 and concluding remarks are given in Section 5.


## 2. Derivation

We propose a BHM for the IVP (2) which can be solved by advancing the from tn to tn+2=tn+2h by fixing two interpolation points {tn,tn+1} and a set of distinct collocation points tn+cih,i=0,1,2,…,10. We then choose the coefficients of the method, such that the method integrates the IVP exactly, whenever the solutions are members of the linear space 1,t,…,t12. Thus, we initially seek a continuous local approximation U(t) on the interval tn,tn+2, which is expressed in vector form as(4)Ut=1tt2t3⋯t12a0a1a2a3⋮a12,where a0,a1,a2,…,a12 are coefficients in Rn to be uniquely determined. To determine these coefficients, we interpolate at tn and tn+1 and then collocate at {tn+cih,i=0,1,…,10}. In particular, we determine these coefficients by imposing the following conditions on (4):(5)Utn=yn,Utn+1=yn+1,U′′tn+cj=fn+cj,j=0,1,…,10,to obtain the following system of equations represented in matrix form(6)1tntn2tn3⋯tn121tn+1tn+12tn+13⋯tn+1120026tn+c0⋯132tn+c0100026tn+c1⋯132tn+c1100026tn+c2⋯132tn+c210⋮⋮⋮⋮⋮0026tn+c10⋯132tn+c1010a0a1a2⋮a12=ynyn+1fn+c0fn+c1fn+c2⋮fn+c10.The system is solved with the aid of Mathematica to obtain the coefficients, a0,a1,a2,…,a12, which are substituted into (4) to obtain the continuous form(7)Ut=1tt2⋯t121tntn2tn3⋯tn121tn+1tn+12tn+13⋯tn+1120026tn+c0⋯132tn+c0100026tn+c1⋯132tn+c1100026tn+c2⋯132tn+c210⋮⋮⋮⋮⋮0026tn+c10⋯132tn+c1010-1ynyn+1fn+c0fn+c1fn+c2⋮fn+c10.The continuous scheme (7) is simplified and evaluated at t=tn+ci,  i=1,…10,i≠5, to give the set of formulas (8), whose coefficients are specified in Tables 1 and 2. In the same vein, the first derivative of the continuous scheme (7) is simplified and evaluated at t=tn+ci,  i=0,1,…10, to give the set of formulas (9), whose coefficients are also specified in Tables 1 and 2.Table 1
Coefficients for (8) and (9).
j c j f n f n + 1 / 5 f n + 2 / 5 f n + 3 / 5 f n + 4 / 5 f n + 1 f n + 6 / 5 f n + 7 / 5 f n + 8 / 5 f n + 9 / 5 f n + 2 y n y n + 1 y n + c j yn+cj′ 0 0 - 2013679 38320128 - 90185 304128 228055 1419264 - 297145 532224 3664715 6386688 - 8081 13824 2553035 6386688 - 310315 1596672 270085 4257792 - 239095 19160064 14299 12773376 - 1 1 0 1   1 1/5 - 761921 399168000 - 442297 13305600 - 21601621 1995840000 - 2016199 49896000 24105979 997920000 - 15717253 498960000 4257371 199584000 - 2593823 249480000 1357603 399168000 - 222953 332640000 120167 1995840000 4 5 1 5 1114177 319334400 - 54021223 2395008000 - 336888491 1596672000 259649 5702400 - 1943911 10368000 51790213 399168000 - 71069639 798336000 8498207 199584000 - 4380451 319334400 236963 88704000 - 1138741 4790016000 - 1 1 1 1   2 2/5 - 5487121 3991680000 - 2183537 79833600 - 9569027 266112000 - 7859413 166320000 2592949 665280000 - 1366229 66528000 1772273 133056000 - 71899 11088000 2825969 1330560000 - 836617 1995840000 150511 3991680000 3 5 2 5 1458827 684288000 23188961 479001600 - 18613933 1596672000 - 1854557 18144000 - 25160417 798336000 - 1811707 399168000 - 275629 159667200 111787 66528000 - 1149229 1596672000 395621 2395008000 - 15703 958003200 - 1 1 1 1   3 3/5 - 3685189 3991680000 - 4015643 221760000 - 3919439 147840000 - 607247 11088000 - 1343147 133056000 - 349817 22176000 6359789 665280000 - 768373 166320000 26767 17740800 - 23731 79833600 35533 1330560000 2 5 3 5 2254403 958003200 11873681 266112000 1152871 16128000 575959 9504000 - 16959871 159667200 6543319 133056000 - 26944583 798336000 3180127 199584000 - 2713189 532224000 473789 479001600 - 140263 1596672000 - 1 1 1 1   4 4/5 - 131239 285120000 - 9055369 997920000 - 5220617 399168000 - 7382783 249480000 - 836489 39916800 - 5019643 498960000 4900639 997920000 - 114739 49896000 1483859 1995840000 - 9703 66528000 5219 399168000 1 5 4 5 3643247 1596672000 9934159 217728000 20360951 319334400 31041793 199584000 26356639 798336000 2988037 399168000 - 10416353 798336000 298957 39916800 - 4206509 1596672000 6857 12672000 - 6833 136857600 - 1 1 1 1   5 5/5 8081 3483648 864845 19160064 853945 12773376 226015 1596672 897065 6386688 412585 3193344 - 237715 6386688 8345 532224 - 61595 12773376 2515 2737152 - 3097 38320128 - 1 1 0 1Table 2
Coefficients for (8) and (9).
j c j f n f n + 1 / 5 f n + 2 / 5 f n + 3 / 5 f n + 4 / 5 f n + 1 f n + 6 / 5 f n + 7 / 5 f n + 8 / 5 f n + 9 / 5 f n + 2 y n y n + 1 y n + c j  yn+cj′ 6 6/5 52559 114048000 57439 6336000 1944059 147840000 46169 1584000 17003551 665280000 4614871 110880000 - 8201 26611200 308353 166320000 - 59651 88704000 39091 285120000 - 16657 1330560000 - 1 5 6 5 2192681 958003200 12112253 266112000 1638893 25344000 284633 1900800 92835103 798336000 33386071 133056000 8008873 114048000 339457 199584000 - 184067 106444800 1030501 2395008000 - 6227 145152000 - 1 1 1 1   7 7/5 525907 570240000 206717 11404800 600379 600379 4153733 71280000 14499731 285120000 2497271 28512000 1779187 57024000 115861 14256000 - 968729 570240000 29903 95040000 - 1753 63360000 - 2 5 7 5 3715013 1596672000 3083923 68428800 107183317 1596672000 28201123 199584000 109363333 798336000 11930899 57024000 33443621 159667200 19286111 199584000 - 15090479 1596672000 11593 8064000 - 109553 958003200 - 1 1 1 1   8 8/5 1378159 997920000 13579127 498960000 13086961 332640000 730607 8316000 2503537 33264000 2237099 16632000 10950581 166320000 652601 13860000 83837 66528000 5647 19958400 - 31249 997920000 - 3 5 8 5 2160553 958003200 109910629 2395008000 100192979 1596672000 4435127 28512000 2394197 22809600 104957957 399168000 107579167 798336000 17260459 66528000 10696153 145152000 - 1127711 479001600 512461 4790016000 - 1 1 1 1   9 9/5 1231639 665280000 1337033 36960000 261313 4928000 1066201 9240000 2304569 22176000 3238111 18480000 11836081 110880000 473339 5544000 2868227 73920000 47953 13305600 - 5219 44352000 - 4 5 9 5 1694713 684288000 11545361 266112000 2687689 35481600 7627681 66528000 153488389 798336000 17118679 133056000 232099897 798336000 405337 3628800 145310747 532224000 164327473 2395008000 - 399227 319334400 - 1 1 1 1   10 10/5 14299 6386688 49025 1064448 396175 6386688 125525 798336 329675 3193344 412585 1596672 329675 3193344 125525 798336 396175 6386688 49025 1064448 14299 6386688 - 1 2 14299 12773376 1121545 19160064 - 17905 12773376 80195 228096 - 1893685 6386688 2691881 3193344 - 273215 580608 1142485 1596672 - 1260145 12773376 243115 709632 2099473 38320128 - 1 1 1 1In order to numerically integrate (1), we advance the BHM from tn to tn+2 on the partition SN, whereby the step [tn,yn,yn′]↦[tn+2,yn+2,yn+2′] is given by combining the following methods:(8)yn+ci=αci,0yn+αci,1yn+1+h2∑j=010βci,cjfn+cj,i=1,…10,i≠5,(9)hyn+ci′=αci,0′yn+αci,1′yn+1+h2∑j=010βci,cj′fn+cj,i=0,…10,where αci,0, αci,1, αci,0′, αci,1′, βci,cj, βci,cj′ are coefficients given in Tables 1 and 2. We note that fn+ci=f(tn+ci,yn+ci,yn+ci′),  y(tn+ci)≈yn+ci, and y′(tn+ci)≈yn+ci′.


## 3. Properties of the Method

Local Truncation Error. We define the local truncation errors (LTEs) of (8) and (9), specified by the coefficients in Tables 1 and 2 as(10)Lciytn;h=ytn+cih-αci,0ytn+αci,1ytn+h+h2∑j=010βci,cjy′′tn+cjh,(11)Lci′ytn;h=hy′tn+cih-αci,0′ytn+αci,1′ytn+h+h2∑j=010βci,cj′y′′tn+cjh.Assuming that y(t) is sufficiently differentiable, we can expand the terms in Lcj and Lcj′ as Taylor series about the point tn to obtain the expressions for the LTEs as follows:LJ[y(tn);h]=Ccjhp+2y(p+2)(tn)+O(hp+3),LJ′[y(tn);h]=Ccj′hp+2y(p+2)(tn)+O(hp+3),wherep is the order and Ccj and Ccj′ are the error constants.Remark 1.
The local truncation error constants of the BHM formulated from (8) and (9) and specified by the coefficients in Tables 1 and 2 as well as choosing ci={0,1/5,…,9/5,2} are displayed in Table 3.Table 3
Coefficients for (8) and (9) including order p and error constant Ccj and Ccj′.
i c i p C c i C c i ′ 0 0 11  - 26927 1330235156250000  1 1/5 11 - 178589 162421875000000000 17123423 4256752500000000000  2 2/5 11 - 477527 974531250000000000 27311 55426464843750000  3 3/5 11 - 911 3806762695312500 31838299 21283762500000000000  4 4/5 11 26927 1330235156250000 3131 3023261718750000  5 1 11  76733 56756700000000000  6 6/5 11 911 3806762695312500 3131 3023261718750000  7 7/5 11 477527 974531250000000000 31838299 21283762500000000000  8 8/5 11 443 634460449218750 27311 55426464843750000  9 9/5 11 178589 162421875000000000 17123423 4256752500000000000  10 2 12 - 200621 1021620600000000000 - 26927 1330235156250000Stability. BHM is formulated from (8) and (9) and defined in vector form as follows:(12)Yμ+1=yn+c1,yn+c2,…,yn+c10,hyn+c1′,hyn+c2′,…,hyn+c10′⊤,Yμ=yn-c9,yn-c8,…,yn-c1,yn,hyn-c9′,hyn-c8′,…,hyn-c1′,hyn′⊤,Fμ+1=fn+c1,fn+c2,…,fn+c10,hfn+c1′,hfn+c2′,…,hfn+c10′⊤,Fμ=fn-c9,fn-c8,…,fn-c1,fn,hfn-c9′,hfn-c8′,…,hfn-c1′,hfn′⊤,where μ=0,…,N,n=0,…,N. The methods in (8) and (9) specified by the coefficients from Tables 1 and 2 are combined to give the BHM, which is expressed as(13)A1Yμ+1=A0Yμ+h2B0Fμ+B1Fμ+1where A0,A1,B0, and B1 are square matrices of dimension twenty whose elements characterize the method and are given by the coefficients of (8) and (9).The linear stability of the BHM is discussed by applying the method to the test equationy′′=-λ2y-δy′, where λ is the frequency and δ is the damping (see [2]). Letting Z=λh and Z∗=δh, it is easily shown as in [2] that the application of (13) to the test equation yields(14)Yμ+1=MZ,Z∗Yμ≔A1-B1Z,Z∗-1A0+B0Z,Z∗Yμ,where the matrix M(Z,Z∗) is the amplification matrix which determines the stability of the method.Definition 2.
The region of stability of the BHM is region throughout which the spectral radiusρ(M(Z,Z∗))≤1 (see [2]).Remark 3.
It is observed that in the(Z,ρ(M(Z)))-plane, the BHM is stable for Z∈[0,59.9896] as demonstrated in example 4.6 and confirmed in Figure 1(b).Figure 1
(a) The stability region for the BHM plotted in the(Z,Z∗)-plane and (b) the stability interval plotted in the (Z,ρ(M(Z)))-plane when δ=0(Z∗=0) for the BHM.
(a) (b)Implementation. The methods specified by (8) and (9) are combined to form the BHM (37), which is then implemented in a block form on the nonoverlapping interval [tn,tn+2],n=0,2,…,N-2. This approach has an advantage over the usual predictor-corrector methods, since it is self-starting. Initially, for n=0, the block method is simultaneously applied on the subinterval [t0,t2] to obtain (yci,yci′)T,i=1,…,10. Using the known values from the previous block, we obtain values on the next subinterval [t2,t4], and the process is repeated until we reach the final interval [tN-2,tN]. In the spirit of [22], our code was written in Mathematica and the steps are given in Algorithm 1.Algorithm 1:BHM algorithm.
( 1 ) procedureEnterPartitions(t0,h,N,tN, variables) ( 2 )  For tn=tn-1+h,n=1,…,N,  h=tN-t0N ( 3 )  Generate block from (13) System      ▹ Initially on [t0,t2] ( 4 )  Solve[System,variables]       ▹ Block-by-block with respect to the variables(solutions) on[tn,tn+2],n=0,2,…,N-2. ( 5 )  Obtain (yn+ci,yn+ci′)T,i=1,…,10,n=0,2,…,N-2. ▹ No need for starting values and predictors ( 6 ) end Procedure


## 4. Numerical Examples

We solve a variety of oscillatory general second-order IVPs and Hamiltonian systems to illustrate the accuracy of our hybrid method. This is done via the maximum global error, where the maximum global error of the approximate solution is calculated asError=max⁡(y(tn)-yn), where y(tn) is the exact solution and yn is the approximate solution on the partition SN. We have also included the maximum global errors for the Hamiltonian (ErrH=max⁡Hn-H0) as well as figures showing the global error of the Hamiltonian (EH=Hn-H0) for each problem.


### 4.1. Application to Hamiltonian Systems

The following methods have been compared in this subsection.(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 in [18].Example 4.
We consider the following oscillatory nonlinear system that was studied in [10]:(15)q′′+13-12-1213q=-∂U∂q,with U(q)=q1q2(q1+q2)3, and initial conditions(16)q0=-11,q′0=-55,such that the analytic solution of this system is q=-cos⁡5t-sin⁡5tcos⁡5t+sin⁡5t.
The Hamiltonian is(17)Hq,q′=12q′Tq′+12qT,13-12-1213q+Uq.This example is solved using both our BHM and GAM and the errors in the solution and Hamiltonian obtained are compared for different step-sizes. The results displayed in Table4 show that our method is more accurate. Details of the results are also displayed in Figure 1.Table 4
Comparison of global error for example 4.1 on the interval[0,100].
h BHM-Error GAM-Error BHM-HE GAM-HE 1/2 4.58 × 1 0 0 1.25 × 1 0 1 2.95 × 1 0 3 5.16 × 1 0 3 1/4 7.54 × 1 0 - 8 7.02 × 1 0 0 3.13 × 1 0 - 8 1.95 × 1 0 4 1/8 2.20 × 1 0 - 11 4.58 × 1 0 0 5.61 × 1 0 - 12 3.37 × 1 0 3 1/16 4.95 × 1 0 - 14 1.95 × 1 0 - 6 1.62 × 1 0 - 12 6.37 × 1 0 - 8 1/32 9.15 × 1 0 - 14 6.66 × 1 0 - 10 3.78 × 1 0 - 12 9.31 × 1 0 - 11Example 5.
We consider the Fermi-Pasta-Ulam problem that has been studied in [10, 25](18)x′′t+Mxt=-∇Ux,t∈0,tend,where(19)M=0m×m0m×m0m×mω2Im×m,Ux=14x1-xm+14+∑i=1m-1xi+1-xm+i-1-xi-xm+i4+xm+x2m4,with initial conditions(20)m=3,ω=50,x10=1,y10=1,x40=1ω,y40=1.
The remaining initial conditions are equal to zero, and the Hamiltonian is(21)Hx,y=12∑i=12myi2+ω22∑i=1mxm+i2+14x1-xm+14+∑i=1m-1xi+1-xm+i-1-xi-xm+i4+xm+x2m4.This example is solved using both our BHM and GAM and the errors in the Hamiltonian obtained are compared for different step-sizes. The results displayed in Table5 show that our method is more accurate. Details of the results are also displayed in Figure 2.Table 5
Comparison of global error for example 4.2 on the interval[0,100].
h BHM-HE GAM-HE 1/8 1.83 × 1 0 - 1 4.06 × 1 0 6 1/16 4.44 × 1 0 - 5 5.02 × 1 0 - 1 1/32 3.03 × 1 0 - 7 1.83 × 1 0 - 1 1/64 2.10 × 1 0 - 12 2.10 × 1 0 - 4 1/128 2.71 × 1 0 - 13 2.16 × 1 0 - 8Figure 2
Results for 4.1. The logarithm of the global error of the Hamiltonian,EH=|Hn-H0| on the interval [0,1000].
(a)
BHM forN=10000 (b)
Adams forN=16000Example 6.
We consider the pendulum oscillator given by(22)q′′=-sin⁡q,with initial conditions(23)q0=0,q′0=1.5,and Hamiltonian(24)H=12q′2-cos⁡q.This example is solved using both our BHM and GAM and the errors in the Hamiltonian obtained are compared for different step-sizes. The results displayed in Table6 show that our method is more accurate. Details of the results are also displayed in Figures 3 and 4.Table 6
Comparison of global error for example 4.3 on the interval[0,50].
h BHM-HE GAM-HE 1/2 7.75 × 1 0 - 9 1.97 × 1 0 - 2 1/4 1.99 × 1 0 - 12 7.31 × 1 0 - 5 1/8 3.24 × 1 0 - 14 1.39 × 1 0 - 7 1/16 7.06 × 1 0 - 14 5.76 × 1 0 - 11 1/32 2.73 × 1 0 - 13 2.25 × 1 0 - 14Figure 3
Results for example 4.2. The logarithm of the global error of the Hamiltonian,EH=|Hn-H0| on the interval [0,1000].
(a)
BHM forN=20000 (b)
Adams forN=40000Figure 4
Results for example 4.3. The logarithm of the global error of the Hamiltonian,EH=Hn-H0 on the interval [0,1000].
(a)
BHM forN=2000 (b)
Adams forN=2000Example 7.
We consider the perturbed Kepler’s problem given by(25)q1′′=-q1q12+q223/2-2ϵ+ϵ2q1q12+q225/2,q10=1,q1′0=0,(26)q2′′=-q2q12+q223/2-2ϵ+ϵ2q2q12+q225/2,q20=0,q2′0=1+ϵ,where the exact solution of this problem is(27)q1t=cos⁡t+ϵt,q2t=sin⁡t+ϵt,and the Hamiltonian is(28)H=12q1′2+q2′2-1q12+q22-2ϵ+ϵ23q12+q223/2.The system also has the angular momentum L=q1q2′-q2q1′ as a first integral. We let ϵ=10-3.This example is solved using both our BHM and GAM and the errors in the solution, Hamiltonian, and momentum obtained are compared for different step-sizes. The results displayed in Table7 show that our method is more accurate. Details of the results are also displayed in Figure 5.Table 7
Comparison of global error for example 4.4 on the interval[0,100].
h BHM-Error GAM-Error BHM-HE GAM-HE BHM-EL GAM-EL 1/2 1.74 × 1 0 - 13 1.86 × 1 0 3 5.27 × 1 0 - 15 4.00 × 1 0 2 5.11 × 1 0 - 15 2.31 × 1 0 0 1/4 8.62 × 1 0 - 13 3.40 × 1 0 4 1.23 × 1 0 - 14 7.35 × 1 0 4 1.24 × 1 0 - 14 2.89 × 1 0 1 1/8 3.49 × 1 0 - 12 1.27 × 1 0 4 2.11 × 1 0 - 14 1.53 × 1 0 4 2.11 × 1 0 - 14 3.07 × 1 0 0 1/16 6.94 × 1 0 - 12 3.78 × 1 0 3 4.19 × 1 0 - 14 8.21 × 1 0 2 4.20 × 1 0 - 14 1.42 × 1 0 0 1/32 1.92 × 1 0 - 12 5.43 × 1 0 - 12 5.06 × 1 0 - 14 3.69 × 1 0 - 14 5.06 × 1 0 - 14 3.69 × 1 0 - 14Figure 5
Results for example 4.3.
(a)
BHM forN=160 (b)
Adams forN=160 (c)
Exact Flow


### 4.2. Oscillatory Problems withy′ Present

The following methods have been compared in this subsection.(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 in [18].Example 8.
We consider the oscillatory system that is solved in [2]:(29)y′′+13-12-1213y=12ε2532-23y′+ε2f1tf2t,y0=ε2ε2,y′0=-46, with f1(t)=36/5sin⁡t+24sin⁡5t and f2(t)=-24/5sin⁡t-36sin⁡5t.
The exact solution of this system isy=sin⁡t-sin⁡5t+εcos⁡tsin⁡t+sin⁡5t+εcos⁡t.This example is solved using both our BHM and GAM and the errors in the solution and the first derivative obtained are compared for different step-sizes. The results displayed in Tables8 and 9 show that our method is more accurate. We note that this example was chosen to show that the BHM performs well on an oscillatory general second-order system of IVPs.Table 8
Comparison of global error for example 4.5 on the interval[0,100].
h BHM-Error(y1) GAM-Error(y1) BHM-Error(y2) GAM-Error(y2) 1 1.57 × 1 0 - 1 1.65 × 1 0 0 1.57 × 1 0 - 1 1.71 × 1 0 0 1/2 9.38 × 1 0 - 5 1.94 × 1 0 0 9.38 × 1 0 - 5 1.94 × 1 0 0 1/4 5.29 × 1 0 - 8 9.88 × 1 0 - 1 5.29 × 1 0 - 8 9.88 × 1 0 - 1 1/8 1.54 × 1 0 - 11 4.72 × 1 0 - 4 1.54 × 1 0 - 11 4.72 × 1 0 - 4 1/16 5.07 × 1 0 - 14 1.36 × 1 0 - 6 5.27 × 1 0 - 14 1.36 × 1 0 - 6Table 9
Comparison of global error of the first derivatives for example 4.5 on the interval[0,100].
h BHM-Error(y1′) GAM-Error(y1′) BHM-Error(y2′) GAM-Error(y2′) 1 7.53 × 1 0 - 1 8.36 × 1 0 0 7.53 × 1 0 - 1 8.35 × 1 0 0 1/2 4.66 × 1 0 - 4 9.82 × 1 0 0 4.66 × 1 0 - 4 9.82 × 1 0 0 1/4 2.61 × 1 0 - 7 4.97 × 1 0 0 2.61 × 1 0 - 7 4.97 × 1 0 0 1/8 7.65 × 1 0 - 11 2.31 × 1 0 - 3 7.64 × 1 0 - 11 2.31 × 1 0 - 3 1/16 9.92 × 1 0 - 14 6.82 × 1 0 - 6 9.84 × 1 0 - 14 6.82 × 1 0 - 6Example 9.
We consider the following general second-order IVP given in [2]:(30)y′′+ω2y=-δy′,(31)y0=1,(32)y′0=-δ2, and the analytical solution is(33)yt=e-δ/2tcostω2-δ24,ω=1,δ=10-3.This example is solved using both our BHM and GAM and the errors obtained are compared for different step-sizes. The results displayed in Table10 show that our method is more accurate.Table 10
Comparison errors for example 4.6 on[0,100].
h BHM-Errors Adams-Errors 2 1.83 × 1 0 - 6 1.51 × 1 0 0 1 7.55 × 1 0 - 10 5.81 × 1 0 - 2 1/2 2.23 × 1 0 - 13 2.97 × 1 0 - 5 1/4 1.77 × 1 0 - 14 2.10 × 1 0 - 8 1/8 4.51 × 1 0 - 14 6.36 × 1 0 - 12


### 4.3. Efficiency Curves for Four Methods

In this subsection, we have solved examples 4.1–4.6 and generated efficiency curves for four methods. In particular, we have compared three methods given in the literature to the BHM given in this paper by plottinglog10(Error) versus N as well as log10(Error) versus CPUTime. The plots given in Figure 7 show that the BHM is the most efficient in terms of the size of N, since smaller values of N give greater accuracy. However, the GAM performs better in terms of CPU time. Nevertheless, the BHM remains competitive pertaining to the CPU time. We note that the high accuracy of the BHM is due to the fact that the error constants for all the members of the BHM are smaller than those of the GAM. For instance, the error constant for the main method for the BHM is -1.964×10-13, while the error constant for the GAM is -5.924×10-3. In this subsection, the following methods have been compared and the details of these comparisons are given in Figure 7:(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 given in [18],(iii)
Block Continuous Hybrid Integrator of order 9 given in [26],(iv)
the 10-step block Falkner-type method of order 11 given in [27].


### 4.4. Application to a PDE

Example 10.
We consider the Telegraph equation which was also solved in [22]:(34)∂2u∂t2+2π∂u∂t+π2u=∂2u∂x2+π2sin⁡πxsin⁡πt+2cos⁡πt,0≤x≤1,0≤t≤1.The analytical solution is given by(35)ux,t=sin⁡πxsin⁡πtand the initial conditions are defined to match the analytical solution.This PDE is solved by first discretization the spatial variablex via the finite difference method to obtain(36)∂2uit∂t2+2π∂uit∂t+π2uit-ui+1t-2uit+ui-1tΔx2=git,0<t<1,uxi,0=ui,utxi,0=ui′,i=1,…,M-1,where Δx=(x0-xM)/M,  xi=x0+iΔx,  i=0,1,…,M,  u=[u1(t),…,uM(t)]T,  G=[g1(t),…,gM(t)]T, ui(t)≈u(xi,t),  u0=0,i=1,…,M,  u′0=πsin⁡(πxi),i=1,…,M, and gi(t)≈g(xi,t)=π2sin⁡πxisin⁡πt+cos⁡πt, which can be written in the form(37)u′′=ft,u,u′,subject to the initial conditions u(t0)=u0,u′t0=u′0, where f(t,u,u′)=Lu+G, andL is an M×M matrix arising from the semidiscretized system and G is a vector of constants (Figure 8).This example is chosen to show that the BHM performs well on PDEs such as the Telegraph equation and this is confirmed by the results displayed in Figure6.Figure 6
Results for example 4.4: the logarithm(log10) of the global error of the Hamiltonian EH=|Hn-H0| and the logarithm of the global error of Momentum EM=|Ln-L0| on the interval [0,1000].
(a)
BHM,N=5000, EH (b)
BHM,N=5000, EM (c)
GAM,N=20000, EH (d)
GAM,N=20000, EMFigure 7
Efficiency curves for examples 4.1-4.6. The labels (a), (c), (e), (g), (i), and (k) represent thelog10(Error) versus N, where N=tN-t0/h. The labels (b), (d), (f), (h), (j), and (l) represent the log10(Error) versus CPUTime.
(a)
Example4 (b)
Example4 (c)
Example5 (d)
Example5 (e)
Example6 (f)
Example6 (g)
Example7 (h)
Example7 (i)
Example8 (j)
Example8 (k)
Example9 (l)
Example9Figure 8
Errors for various values of N and M for example 4.7.
(a)
N = M = 16 (b)
N = M = 32


### 4.5. Application to a Stiff Problem

Example 11.
We consider the stiff second-order IVP (see [28])(38)y1′′=ε-2y1+2ε-2y2,(39)y2′′=1-εy1+1-2εy2,(40)y10=2,(41)y1′0=0,(42)y20=-1,(43)y2′0=0,(44)ε=2500,(45)tϵ0,100.y1(t)=2cos⁡t,y2(t)=-cost, where ε is an arbitrary parameter.This problem was chosen to justify that the stability of the BHM is achieved whenZ∈[0,59.9896]. In Table 11, we give the absolute errors at selected values of t, which indicate that, choosing N=646, the method is stable since, for this value of N, Z∈[0,59.9896]. However, for N=645, Z∋[0,59.9896]; hence the method becomes unstable.Table 11
Demonstrating stability using absolute errors for example 4.7. on the interval[0,100].
x N = 646 Z ∈ 0,59.9896 N = 645 Z ∋ 0,59.9896 BHM-Err BHM-Err 10.0 1.4 × 1 0 - 13 3.2 × 1 0 - 12 20.0 1.0 × 1 0 - 12 2.9 × 1 0 - 11 30.0 2.1 × 1 0 - 12 4.0 × 1 0 - 11 40.0 2.5 × 1 0 - 12 1.7 × 1 0 - 9 50.0 4.1 × 1 0 - 14 2.0 × 1 0 - 7 60.0 2.8 × 1 0 - 12 4.0 × 1 0 - 6 70.0 1.8 × 1 0 - 12 6.4 × 1 0 - 6 80.0 7.9 × 1 0 - 14 2.6 × 1 0 - 4 90.0 1.4 × 1 0 - 12 3.2 × 1 0 - 2 100.0 1.7 × 1 0 - 12 6.3 × 1 0 - 1


## 4.1. Application to Hamiltonian Systems

The following methods have been compared in this subsection.(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 in [18].Example 4.
We consider the following oscillatory nonlinear system that was studied in [10]:(15)q′′+13-12-1213q=-∂U∂q,with U(q)=q1q2(q1+q2)3, and initial conditions(16)q0=-11,q′0=-55,such that the analytic solution of this system is q=-cos⁡5t-sin⁡5tcos⁡5t+sin⁡5t.
The Hamiltonian is(17)Hq,q′=12q′Tq′+12qT,13-12-1213q+Uq.This example is solved using both our BHM and GAM and the errors in the solution and Hamiltonian obtained are compared for different step-sizes. The results displayed in Table4 show that our method is more accurate. Details of the results are also displayed in Figure 1.Table 4
Comparison of global error for example 4.1 on the interval[0,100].
h BHM-Error GAM-Error BHM-HE GAM-HE 1/2 4.58 × 1 0 0 1.25 × 1 0 1 2.95 × 1 0 3 5.16 × 1 0 3 1/4 7.54 × 1 0 - 8 7.02 × 1 0 0 3.13 × 1 0 - 8 1.95 × 1 0 4 1/8 2.20 × 1 0 - 11 4.58 × 1 0 0 5.61 × 1 0 - 12 3.37 × 1 0 3 1/16 4.95 × 1 0 - 14 1.95 × 1 0 - 6 1.62 × 1 0 - 12 6.37 × 1 0 - 8 1/32 9.15 × 1 0 - 14 6.66 × 1 0 - 10 3.78 × 1 0 - 12 9.31 × 1 0 - 11Example 5.
We consider the Fermi-Pasta-Ulam problem that has been studied in [10, 25](18)x′′t+Mxt=-∇Ux,t∈0,tend,where(19)M=0m×m0m×m0m×mω2Im×m,Ux=14x1-xm+14+∑i=1m-1xi+1-xm+i-1-xi-xm+i4+xm+x2m4,with initial conditions(20)m=3,ω=50,x10=1,y10=1,x40=1ω,y40=1.
The remaining initial conditions are equal to zero, and the Hamiltonian is(21)Hx,y=12∑i=12myi2+ω22∑i=1mxm+i2+14x1-xm+14+∑i=1m-1xi+1-xm+i-1-xi-xm+i4+xm+x2m4.This example is solved using both our BHM and GAM and the errors in the Hamiltonian obtained are compared for different step-sizes. The results displayed in Table5 show that our method is more accurate. Details of the results are also displayed in Figure 2.Table 5
Comparison of global error for example 4.2 on the interval[0,100].
h BHM-HE GAM-HE 1/8 1.83 × 1 0 - 1 4.06 × 1 0 6 1/16 4.44 × 1 0 - 5 5.02 × 1 0 - 1 1/32 3.03 × 1 0 - 7 1.83 × 1 0 - 1 1/64 2.10 × 1 0 - 12 2.10 × 1 0 - 4 1/128 2.71 × 1 0 - 13 2.16 × 1 0 - 8Figure 2
Results for 4.1. The logarithm of the global error of the Hamiltonian,EH=|Hn-H0| on the interval [0,1000].
(a)
BHM forN=10000 (b)
Adams forN=16000Example 6.
We consider the pendulum oscillator given by(22)q′′=-sin⁡q,with initial conditions(23)q0=0,q′0=1.5,and Hamiltonian(24)H=12q′2-cos⁡q.This example is solved using both our BHM and GAM and the errors in the Hamiltonian obtained are compared for different step-sizes. The results displayed in Table6 show that our method is more accurate. Details of the results are also displayed in Figures 3 and 4.Table 6
Comparison of global error for example 4.3 on the interval[0,50].
h BHM-HE GAM-HE 1/2 7.75 × 1 0 - 9 1.97 × 1 0 - 2 1/4 1.99 × 1 0 - 12 7.31 × 1 0 - 5 1/8 3.24 × 1 0 - 14 1.39 × 1 0 - 7 1/16 7.06 × 1 0 - 14 5.76 × 1 0 - 11 1/32 2.73 × 1 0 - 13 2.25 × 1 0 - 14Figure 3
Results for example 4.2. The logarithm of the global error of the Hamiltonian,EH=|Hn-H0| on the interval [0,1000].
(a)
BHM forN=20000 (b)
Adams forN=40000Figure 4
Results for example 4.3. The logarithm of the global error of the Hamiltonian,EH=Hn-H0 on the interval [0,1000].
(a)
BHM forN=2000 (b)
Adams forN=2000Example 7.
We consider the perturbed Kepler’s problem given by(25)q1′′=-q1q12+q223/2-2ϵ+ϵ2q1q12+q225/2,q10=1,q1′0=0,(26)q2′′=-q2q12+q223/2-2ϵ+ϵ2q2q12+q225/2,q20=0,q2′0=1+ϵ,where the exact solution of this problem is(27)q1t=cos⁡t+ϵt,q2t=sin⁡t+ϵt,and the Hamiltonian is(28)H=12q1′2+q2′2-1q12+q22-2ϵ+ϵ23q12+q223/2.The system also has the angular momentum L=q1q2′-q2q1′ as a first integral. We let ϵ=10-3.This example is solved using both our BHM and GAM and the errors in the solution, Hamiltonian, and momentum obtained are compared for different step-sizes. The results displayed in Table7 show that our method is more accurate. Details of the results are also displayed in Figure 5.Table 7
Comparison of global error for example 4.4 on the interval[0,100].
h BHM-Error GAM-Error BHM-HE GAM-HE BHM-EL GAM-EL 1/2 1.74 × 1 0 - 13 1.86 × 1 0 3 5.27 × 1 0 - 15 4.00 × 1 0 2 5.11 × 1 0 - 15 2.31 × 1 0 0 1/4 8.62 × 1 0 - 13 3.40 × 1 0 4 1.23 × 1 0 - 14 7.35 × 1 0 4 1.24 × 1 0 - 14 2.89 × 1 0 1 1/8 3.49 × 1 0 - 12 1.27 × 1 0 4 2.11 × 1 0 - 14 1.53 × 1 0 4 2.11 × 1 0 - 14 3.07 × 1 0 0 1/16 6.94 × 1 0 - 12 3.78 × 1 0 3 4.19 × 1 0 - 14 8.21 × 1 0 2 4.20 × 1 0 - 14 1.42 × 1 0 0 1/32 1.92 × 1 0 - 12 5.43 × 1 0 - 12 5.06 × 1 0 - 14 3.69 × 1 0 - 14 5.06 × 1 0 - 14 3.69 × 1 0 - 14Figure 5
Results for example 4.3.
(a)
BHM forN=160 (b)
Adams forN=160 (c)
Exact Flow


## 4.2. Oscillatory Problems withy′ Present

The following methods have been compared in this subsection.(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 in [18].Example 8.
We consider the oscillatory system that is solved in [2]:(29)y′′+13-12-1213y=12ε2532-23y′+ε2f1tf2t,y0=ε2ε2,y′0=-46, with f1(t)=36/5sin⁡t+24sin⁡5t and f2(t)=-24/5sin⁡t-36sin⁡5t.
The exact solution of this system isy=sin⁡t-sin⁡5t+εcos⁡tsin⁡t+sin⁡5t+εcos⁡t.This example is solved using both our BHM and GAM and the errors in the solution and the first derivative obtained are compared for different step-sizes. The results displayed in Tables8 and 9 show that our method is more accurate. We note that this example was chosen to show that the BHM performs well on an oscillatory general second-order system of IVPs.Table 8
Comparison of global error for example 4.5 on the interval[0,100].
h BHM-Error(y1) GAM-Error(y1) BHM-Error(y2) GAM-Error(y2) 1 1.57 × 1 0 - 1 1.65 × 1 0 0 1.57 × 1 0 - 1 1.71 × 1 0 0 1/2 9.38 × 1 0 - 5 1.94 × 1 0 0 9.38 × 1 0 - 5 1.94 × 1 0 0 1/4 5.29 × 1 0 - 8 9.88 × 1 0 - 1 5.29 × 1 0 - 8 9.88 × 1 0 - 1 1/8 1.54 × 1 0 - 11 4.72 × 1 0 - 4 1.54 × 1 0 - 11 4.72 × 1 0 - 4 1/16 5.07 × 1 0 - 14 1.36 × 1 0 - 6 5.27 × 1 0 - 14 1.36 × 1 0 - 6Table 9
Comparison of global error of the first derivatives for example 4.5 on the interval[0,100].
h BHM-Error(y1′) GAM-Error(y1′) BHM-Error(y2′) GAM-Error(y2′) 1 7.53 × 1 0 - 1 8.36 × 1 0 0 7.53 × 1 0 - 1 8.35 × 1 0 0 1/2 4.66 × 1 0 - 4 9.82 × 1 0 0 4.66 × 1 0 - 4 9.82 × 1 0 0 1/4 2.61 × 1 0 - 7 4.97 × 1 0 0 2.61 × 1 0 - 7 4.97 × 1 0 0 1/8 7.65 × 1 0 - 11 2.31 × 1 0 - 3 7.64 × 1 0 - 11 2.31 × 1 0 - 3 1/16 9.92 × 1 0 - 14 6.82 × 1 0 - 6 9.84 × 1 0 - 14 6.82 × 1 0 - 6Example 9.
We consider the following general second-order IVP given in [2]:(30)y′′+ω2y=-δy′,(31)y0=1,(32)y′0=-δ2, and the analytical solution is(33)yt=e-δ/2tcostω2-δ24,ω=1,δ=10-3.This example is solved using both our BHM and GAM and the errors obtained are compared for different step-sizes. The results displayed in Table10 show that our method is more accurate.Table 10
Comparison errors for example 4.6 on[0,100].
h BHM-Errors Adams-Errors 2 1.83 × 1 0 - 6 1.51 × 1 0 0 1 7.55 × 1 0 - 10 5.81 × 1 0 - 2 1/2 2.23 × 1 0 - 13 2.97 × 1 0 - 5 1/4 1.77 × 1 0 - 14 2.10 × 1 0 - 8 1/8 4.51 × 1 0 - 14 6.36 × 1 0 - 12


## 4.3. Efficiency Curves for Four Methods

In this subsection, we have solved examples 4.1–4.6 and generated efficiency curves for four methods. In particular, we have compared three methods given in the literature to the BHM given in this paper by plottinglog10(Error) versus N as well as log10(Error) versus CPUTime. The plots given in Figure 7 show that the BHM is the most efficient in terms of the size of N, since smaller values of N give greater accuracy. However, the GAM performs better in terms of CPU time. Nevertheless, the BHM remains competitive pertaining to the CPU time. We note that the high accuracy of the BHM is due to the fact that the error constants for all the members of the BHM are smaller than those of the GAM. For instance, the error constant for the main method for the BHM is -1.964×10-13, while the error constant for the GAM is -5.924×10-3. In this subsection, the following methods have been compared and the details of these comparisons are given in Figure 7:(i)
Block Hybrid Method (BHM) of order 11 given in this paper,(ii)
Generalized Adams method (GAM) of order 11 given in [18],(iii)
Block Continuous Hybrid Integrator of order 9 given in [26],(iv)
the 10-step block Falkner-type method of order 11 given in [27].


## 4.4. Application to a PDE

Example 10.
We consider the Telegraph equation which was also solved in [22]:(34)∂2u∂t2+2π∂u∂t+π2u=∂2u∂x2+π2sin⁡πxsin⁡πt+2cos⁡πt,0≤x≤1,0≤t≤1.The analytical solution is given by(35)ux,t=sin⁡πxsin⁡πtand the initial conditions are defined to match the analytical solution.This PDE is solved by first discretization the spatial variablex via the finite difference method to obtain(36)∂2uit∂t2+2π∂uit∂t+π2uit-ui+1t-2uit+ui-1tΔx2=git,0<t<1,uxi,0=ui,utxi,0=ui′,i=1,…,M-1,where Δx=(x0-xM)/M,  xi=x0+iΔx,  i=0,1,…,M,  u=[u1(t),…,uM(t)]T,  G=[g1(t),…,gM(t)]T, ui(t)≈u(xi,t),  u0=0,i=1,…,M,  u′0=πsin⁡(πxi),i=1,…,M, and gi(t)≈g(xi,t)=π2sin⁡πxisin⁡πt+cos⁡πt, which can be written in the form(37)u′′=ft,u,u′,subject to the initial conditions u(t0)=u0,u′t0=u′0, where f(t,u,u′)=Lu+G, andL is an M×M matrix arising from the semidiscretized system and G is a vector of constants (Figure 8).This example is chosen to show that the BHM performs well on PDEs such as the Telegraph equation and this is confirmed by the results displayed in Figure6.Figure 6
Results for example 4.4: the logarithm(log10) of the global error of the Hamiltonian EH=|Hn-H0| and the logarithm of the global error of Momentum EM=|Ln-L0| on the interval [0,1000].
(a)
BHM,N=5000, EH (b)
BHM,N=5000, EM (c)
GAM,N=20000, EH (d)
GAM,N=20000, EMFigure 7
Efficiency curves for examples 4.1-4.6. The labels (a), (c), (e), (g), (i), and (k) represent thelog10(Error) versus N, where N=tN-t0/h. The labels (b), (d), (f), (h), (j), and (l) represent the log10(Error) versus CPUTime.
(a)
Example4 (b)
Example4 (c)
Example5 (d)
Example5 (e)
Example6 (f)
Example6 (g)
Example7 (h)
Example7 (i)
Example8 (j)
Example8 (k)
Example9 (l)
Example9Figure 8
Errors for various values of N and M for example 4.7.
(a)
N = M = 16 (b)
N = M = 32


## 4.5. Application to a Stiff Problem

Example 11.
We consider the stiff second-order IVP (see [28])(38)y1′′=ε-2y1+2ε-2y2,(39)y2′′=1-εy1+1-2εy2,(40)y10=2,(41)y1′0=0,(42)y20=-1,(43)y2′0=0,(44)ε=2500,(45)tϵ0,100.y1(t)=2cos⁡t,y2(t)=-cost, where ε is an arbitrary parameter.This problem was chosen to justify that the stability of the BHM is achieved whenZ∈[0,59.9896]. In Table 11, we give the absolute errors at selected values of t, which indicate that, choosing N=646, the method is stable since, for this value of N, Z∈[0,59.9896]. However, for N=645, Z∋[0,59.9896]; hence the method becomes unstable.Table 11
Demonstrating stability using absolute errors for example 4.7. on the interval[0,100].
x N = 646 Z ∈ 0,59.9896 N = 645 Z ∋ 0,59.9896 BHM-Err BHM-Err 10.0 1.4 × 1 0 - 13 3.2 × 1 0 - 12 20.0 1.0 × 1 0 - 12 2.9 × 1 0 - 11 30.0 2.1 × 1 0 - 12 4.0 × 1 0 - 11 40.0 2.5 × 1 0 - 12 1.7 × 1 0 - 9 50.0 4.1 × 1 0 - 14 2.0 × 1 0 - 7 60.0 2.8 × 1 0 - 12 4.0 × 1 0 - 6 70.0 1.8 × 1 0 - 12 6.4 × 1 0 - 6 80.0 7.9 × 1 0 - 14 2.6 × 1 0 - 4 90.0 1.4 × 1 0 - 12 3.2 × 1 0 - 2 100.0 1.7 × 1 0 - 12 6.3 × 1 0 - 1


## 5. Conclusion

We have proposed a BHM of order 11 for solving systems of general second-order IVPs and Hamiltonian systems. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and implemented in a block-by-block manner. In this way, starting values and predictors which are generally considered drawbacks in the implementation of predictor-corrector methods are avoided. The properties of the BHM are discussed and the performance of the method is demonstrated on several numerical examples. In particular, the superiority in terms of accuracy of the BHM over comparable methods given in the literature is established numerically. Our future research will be focused on developing methods for oscillatory systems arising from the semidiscretization of hyperbolic partial differential equations.



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