High Order Block Implicit Multi-step (HOBIM) Method For the Solution of Stiff Ordinary Differential Equation

ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url:http://www.ijpam.eu

doi:http://dx.doi.org/10.12732/ijpam.v96i4.5

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HIGH ORDER BLOCK IMPLICIT MULTI-STEP (HOBIM) METHODS FOR THE SOLUTION OF STIFF

ORDINARY DIFFERENTIAL EQUATIONS

J.P. Chollom^1§, G.M. Kumleng², S. Longwap³

1,2,3Department of Mathematics University of Jos

Jos, NIGERIA

Abstract: The search for higher order A-stable linear multi-step methods has been the interest of many numerical analyst and has been realized through either higher derivatives of the solution or by inserting additional off step points,supper future points and the likes.These methods are suitable for the solution of stiff differential equations which exhibit characteristics that place severe restriction on the choice of step size. It becomes necessary that only methods with large regions of absolute stability remain suitable for such equa- tions. In this paper, high order block implicit multi-step methods of the hybrid form up to order twelve have been constructed using the multi-step collocation approach by inserting one or more off step points in the multi-step method.

The accuracy and stability properties of the new methods are investigated and are shown to yield A- stable methods, a property desirable of methods suitable for the solution of stiff ODEs. The new High Order Block Implicit Multistep methods used as block integrators are tested on stiff differential systems and the results reveal that the new methods are efficient and compete favorably with the state of the art Matlab ode23 code.

AMS Subject Classification: 34A45, 65L06

Key Words: block linear integrators, multi step collocation, A-stability, stiff systems, chemical reactions

Received: April 6, 2014 ^c 2014 Academic Publications, Ltd.

url: www.acadpubl.eu

§Correspondence author

484 J.P. Chollom, G.M. Kumleng, S. Longwap 1. Introduction

Stiff differential equations have been known to cause singular computational difficulties because of the severe restriction on the step size used. To solve the stiff ODE (1), various authors have made several attempts and came up with various methods of solution.

y^′ =f(x, y), y(0) = 0. (1)

The traditional approach has been the Adams code developed by Shampine [28]

where the Adams Moulton formula was used as predictor and Adams Bashfort formula as corrector. Though these yielded a very successful combination but have a draw back because of its requirements for starting values which could lead to growing numerical errors and corrupting further approximations May- ers and Suli[24]. To resolve the issue of starting value, Onimanyi etal[25],[26]

proposed the block Linear Multistep methods based on the multi-step colloca- tion approach of Lie and Norset [23] and the self-starting methods of Cash [6].

These methods were developed through the continuous formulation of the linear k-step methods which provided sufficient number of simultaneous discrete meth- ods used as single integrators. However most of these methods were not able to handle stiff ODEs due to stability issues. A popular approach that worked for this class of methods was that by Gear [17],Ascher and Petzold [2] who devel- oped the Backward Differentiation formula of step number k and order k with distinguish features where f is evaluated at the current step [xn, y_n] only. The k-step Adams Moulton methods have failed to perform successfully on stiff equa- tions . Researchers then resorted to implicit Runge-Kutta (IRK) methods, the Backward Differentiation Formulae (BDF) and recently the collocation meth- ods (Hairer and Wanner [19] and Enright [13].These methods have been useful in handling stiff equations due to their better stability properties. To satisfy the A-stability property,the following researchers, Enright [13], Cash [7], [8], [9], Lie and Norsett [23], Chollom [11],Okuonghae and Ikhile [27],Akinfenwa,et al [3] and Ezzeddine and Hojjati [14] developed numerical methods that are A-stable and very suitable for the solution of stiff equations. In this paper, we pursue the hybrid block approach of Chollom and Onumanyi [12] were they constructed block hybrid Adams Moulton methods for 1≤ K ≤ 5 which pro- duced methods that are shown to possess better stability properties than single integrators and suitable for stiff ODE’s.The paper constructed high order block implicit multi-step (HOBIM) methods for 6≤K ≤10 to advance the integra- tion forward.. The rest of the paper is divided as follows: The derivation of the new methods is done in Section 2, the convergence analysis is in Section

3, Numerical experiment to test the efficiency of the new methods is done in Section 4 and the concluding remarks in Section 5

2. Derivation of the HOBIM Methods

The high order block implicit multi-step (HOBIM) methods are derived from a class of the multi-step collocation methods with continuous coefficients of the Adams class.These methods in their continuous form are expressed as:

y(x) =

k

X

j=0

ψjy_n+j−h

k

X

j=0

βjf_n+j (2)

forkthe step numberk >0,h a constant step size given byh=x_n+r−x_n, r= 1,2, ..., k.The approximation (2) is formulated into the generalization

y(x) =

t−1

X

j=0

ψ_jy_n+j =h

s−1

X

j=0

β_jf(x, y(x)) (3)

And is defined over the interval x∈(x_n+k−1, x_n+k), where Ψ_j(x) =

r+m−1

X

j=0

α_j,j+1x^jhβ_j(x) =h

r+m−1

X

j=0

β_j,j+1x^j (4)

are the continuous coefficients to be determined satisfying the following inter- polation and collocation conditions.

y(x_n+j) =y_n+j, j ∈(0,1, . . . , r−1)

y^′(x_j) =f_n+j, j ∈(0,1, . . . , m−1) (5) wheref_n+j =f(x_n+j, y_n+j).The interpolation and collocation conditions being r−1 andm−1 respectively .From equations (4) and (5) we have the following imposed conditions:

αjx_n+j =δij, j∈(0,1, . . . , r−1), i∈(0,1, . . . , r−1) hβ_jx_n+j = 0, j∈(0,1, . . . , r−1), i∈(0,1, . . . , r−1) α^′_jx_j = 0, j ∈(0,1, . . . , r−1), i∈(0,1, . . . , r−1) hβ_j^′x_n+j =δ_ij, j ∈(0,1, . . . , m−1), i∈(0,1, . . . , r−1)

(6)

Expressing (6) in matrix form yields the equation

DC=I (7)

486 J.P. Chollom, G.M. Kumleng, S. Longwap I being an identity matrix of orderr×m and the matrix D given by

D=

























1 x_n x²_n . . . x^t_n . . . x^t+s−1_n 1 x_n+1 x²_n+1 . . . x^t_n+1 . . . x^t+s−1_n+1

... ... ... ... ... ... ... 1 x_n+k−1 x²_n+k−1 . . . x^t_n+k−1 . . . x^t+s−1_n+k−1 0 1 2xn . . . (t)xn . . . (t+s−1)xn

... ... ... ... ... ... ...

0 1 2x_s−1 . . . (t)x^t−1_s−1 . . . (t+s−1)x^(t+s−2)_s−1























 (8)

is a non-singular matrix of dimension (s+t)×(s+t).The matrix C given in (9) and is of

C=











α₀₁ α₁₁ . . . α_t−1,1 hβ₀₁ . . . hβ_s−1,1 α₀₂ α₁₂ . . . α_t−1,2 hβ₀₂ . . . hβ_s−1,2

... ... ... ... ... ... ... α_0,t+s α_1,t+s . . . α_t−1,t+s hβ_0,t+s . . . hβ_s−1,t+s











(9)

dimension (s+t)× (s+t) defined byC=D⁻¹ =C_ij, i, j = 1,. . ., s+t−1 The entries of the matrixC substituted into (4) produces the continuous coefficients of the method referred to as the continuous formulation of the Adams Moulton Class. The continuous interpolant evaluated at both grid and off grid points results in the methods discrete schemes used as block integrators.

2.1. Derivation of HOBIM k= 6

Consideringk= 6, x∈[xn, x_n+6] .The continuous form of the HOBIM method k= 6 is given by

y(x) =

k

X

j=0

ψ_j(x)y_n+j +h

k

X

j=0

β_j(x)f_n+j+β_j(µ), µ= 11

2 (10)

Evaluating the matrix (8) with s = 7, t = 2 produces the matrix of the method D in (11)

D=















1 x_n+5 x²_n+5 . . . x⁸_n+5 0 1 2x_n . . . 8x⁷_n

... ... ... ... ... 0 1 2x_n+µ . . . 8x⁷_n+µ 0 1 2x_n+6 . . . 8x⁷_n+6















(11)

Inverting (11) in a Maple environment yields the elements of the matrix C in (9).From the elements of C we obtain the continuous coefficients of the HOBIM method for k= 6 in (12).

α₅(x) = 1

hβ₀(x) = ^579ζ_440h² +^1337ζ_1485h³2 −_31680h^11333ζ43 +ζ+ _7920h^679ζ54 −_47520h^581ζ65 −_1008h²⁹⁵ −_55440h^53ζ75

−_31680h^ζ8 7

hβ₁(x) = ^11ζ_3h² −^359ζ_90h2³ + ^2117ζ_1080h⁴3 −_360h^191ζ54 +_648h^53ζ65 −_18144h²⁸⁰²⁵ −_2520h^17ζ75 −_4320h^ζ8 7

hβ₂(x) =−^165ζ_28h² +^67ζ_8h2³ −_192h^925ζ43 + ^347ζ_240h⁵4 −^23ζ_96h⁶5 − _672h¹²⁵ −_48h^ζ75 −_1344h^ζ8 7

hβ₃(x) = ^22ζ_3h² −^1517ζ_135h2³ +^1277ζ_180h3⁴ − ^83ζ_36h⁵4 +_27h^11ζ65 −_1008h¹⁹⁷⁵ − _1260h^47ζ75 −_720h^ζ87

hβ₄(x) = ^55ζ_8h² −^131ζ_12h2³ − ^4169ζ_576h3⁴ +^199ζ_80h4⁵ −_864h^401ζ65 − _756h¹²⁵ + _112h^5ζ75 −_576h^ζ87

hβ₅(x) = ^33ζ_5h² −^107ζ_10h2³ + ^877ζ_120h⁴3 −^313ζ_120h⁵4 +_120h^61ζ65 −_972h⁹⁹⁵ −_840h^43ζ75 +_489h^ζ87

hβ¹¹

2 (x) = ^55ζ_8h² − ^131ζ_12h2³ −^4169ζ_576h3⁴ + ^199ζ_80h4⁵ −^401ζ_864h⁶5 −_756h¹²⁵ +_112h^5ζ75 −_576h^ζ87

hβ₆(x) = ^55ζ_8h² −^131ζ_12h2³ − ^4169ζ_576h3⁴ +^199ζ_80h4⁵ −_864h^401ζ65 − _756h¹²⁵ + _112h^5ζ75 −_576h^ζ87

(12) Substituting (12) into (10) with µ = ¹¹₂,ζ = x−xn produces the continuous form of the multistep collocation method for K= 6 as:

¯

y(x) =y_n+5+ [−_440h⁵⁷⁹ ζ²+_1485h¹³³⁷2ζ³−_31680h¹¹³³3ζ⁴+ζ+_7920h⁶⁷⁹4ζ⁵−_47520h⁵⁸¹ 5ζ⁶

−_1008h²⁹⁵ −_55440h⁵³ 6ζ⁷

−_31680h¹ 7ζ⁸]fn+ [¹¹_3hζ²− _90h³⁵⁹2ζ³+_1080h²¹¹⁷3ζ⁴−_360h¹⁹¹4ζ⁵+ _648h⁵³5ζ⁶−_18144h²⁸⁰²⁵

−_2520h¹⁷ 6ζ⁷

−_4320h¹ 7ζ⁸]f_n+ 1 + [−¹⁶⁵_28hζ^{+ 67}_8h2ζ³−_192h⁹²⁵3ζ⁴+ _240h³⁴⁷4ζ^96h235ζ⁶− _672h¹²⁵ +_48h¹6ζ⁷

−_1344h¹ 7ζ⁸]f_n+2 +[²²_3hζ²−_135h¹⁵¹⁷2ζ³+_180h¹²⁷⁷3ζ⁴−_36h⁸³4ζ⁵+_27h¹¹5ζ⁶−_1008h¹⁹⁷⁵ − _1260h⁴⁷ 6ζ⁷+_720h¹ 7ζ⁸]f_n+3 +[−_8h⁵⁵ζ²+_12h¹³¹2ζ³−_576h⁴¹⁶⁹3ζ⁴+_80h¹⁹⁹4ζ⁵−_864h⁴⁰¹5ζ⁶−_756h¹²⁵ +_112h⁵ 6ζ⁷

−_576h¹ ₇ζ⁸]f_n+4+ [³³_5hζ²

−_10h¹⁰⁷2ζ³+_120h⁸⁷⁷3ζ⁴−_120h³¹³4ζ⁵+ _120h⁶¹5ζ⁶−_972h⁹⁹⁵ −_840h⁴³6ζ⁷+_489h¹ 7ζ⁸]f_n+5

+ [_12h¹¹ζ²−_1080h¹⁶²⁷2ζ³ (13) +_2880h³⁰²³3ζ⁴−_720h²²⁷4ζ⁵+_864h⁶⁷5ζ⁶−_2016h²⁷⁵ −_5040h⁴¹ 6ζ⁷+_2880h¹ 7ζ⁸]f_n+11

2 + [_5h³³ζ²

− _10h¹⁰⁷2ζ³ +_120h⁸⁷⁷3ζ⁴−_120h³¹³4ζ⁵+_120h⁶¹5ζ⁶− _756h⁹⁹⁵ − _840h⁴³6ζ⁷+_480h¹ 7ζ⁸]f_n+6

488 J.P. Chollom, G.M. Kumleng, S. Longwap Table 1: Coefficients of HOBIM k= 6

k 1 2 3 4 5 ¹¹₂ 6

α₀ 0 0 0 0 1 0 0

α₅ 1 1 1 1 0 1 1

β_{0 945}⁸ -₃₆₉₆₀⁷⁸ ₁₂₄₇₄⁴⁵ −₃₃₂₆₄₀²⁶² ₁₈₁₄₄⁵³⁰⁰ -_15482880²³³⁵ ₉₀₇₂₀¹⁸ β₁ -³⁴²_{945 36960}⁹⁷⁸⁹ -₁₂₄₇₄⁵⁵⁰ ₃₃₂₆₄₀^{2475 28025}_{18144 15482880}²¹¹⁵⁰ -₉₀₇₂₀¹⁵⁷ β₂ -¹²²⁴₉₄₅ -¹⁵⁹³⁹_{36960 12474}³⁸⁶¹ -₃₃₂₆₄₀¹¹¹⁸⁷ ₁₈₁₄₄³³⁷⁵ -_15482880⁸⁸⁸⁹³ ₉₀₇₂₀⁶²¹ β₃ -⁶⁶⁴¹₉₄₅ -⁴¹⁹¹⁰₃₆₉₆₀ -⁵³⁰⁶⁴_{12474 332640}^{35354 35550}_{18144 15482880}²³⁹⁷³² -₉₀₇₂₀¹⁴⁹⁴ β₄ -¹²²⁴₉₄₅ -³⁵⁹⁰⁴₃₆₉₆₀ -¹⁴⁸⁹²⁹₁₂₄₇₄ -¹⁹⁰⁸⁷²_{332640 18144}³⁰⁰ -_15482880⁵⁴⁰⁸⁷³ ₉₀₇₂₀²⁴⁹⁶ β₅ -³⁴²₉₄₅ -₃₆₉₆₀²³⁶⁶ -₁₂₄₇₄⁹⁹⁹⁴ -²¹⁹²⁸⁵₃₃₂₆₄₀ ⁵⁸⁷³⁵_{18144 15482880}^{4566222 11043}₉₀₇₂₀ β11

2 0 ₃₆₉₆₀^{6656 10240}_{12474 332640}⁵⁹⁹⁰⁴ -¹²⁸⁰⁰_{18144 15482880}^{3732480 64000}₉₀₇₂₀ β_{6 945}⁸ -₃₆₉₆₀¹⁰²³ -₁₂₄₇₄¹⁰⁸⁹ -₃₃₂₆₄₀⁸⁷⁶ ₁₈₁₄₄²⁴⁷⁵ -_15482880^{186043 14103}₉₀₇₂₀

Evaluating (13) atζ = 0, h,2h,3h,4h,¹¹₂ ,and 6h yields the coefficients of the discrete HOBIM method for k = 6 used used as block integrator given in Table1.

2.2. Derivation of HOBIM k= 7

Consideringk= 7, x∈[x_n, x_n+7] The continuous form of the method HOBIM k=7 is given by

y(x) =

k

X

j=0

ψ_j(x)y_n+j +h

k

X

j=0

β_j(x)f_n+j+β_j(µ), µ= 13

2 (14)

Evaluating the matrix (8) with s = 8, t = 2 produces the matrix of the method D in (15)

D=















1 x_n+6 x²_n+6 . . . x⁹_n+6 0 1 2x_n . . . 9x⁸_n

... ... ... ... ... 0 1 2x_n+µ . . . 9x⁸_n+µ 0 1 2x_n+7 . . . 9x⁸_n+7















(15)

Inverting (15) in a Maple environment yields the elements of the matrixC .Substituting the elements ofC into (4) produces the continuous coefficients of the method which are substituted into (14) to give the continuous form of the multistep collocation method (16).

y(x) =y_n+6+[− 4999

3640hζ²+49213

49140h²ζ³− 5441

12480h³ζ⁴+ζ− 929

7800h⁴ζ⁵− 193 9360h⁵ζ⁶

− 2593

9100h − 1

455h⁶ζ⁷− 23

17420h⁷ζ⁸+ 1

294840h⁸ζ⁹]fn+ [− 91

22hζ²− 20 60h²ζ³ + 10301

3960h³ζ⁴+ζ−31853

39600h⁴ζ⁵+ 1433

9504h⁵ζ⁶− 3096

1925h − 941 5440h⁶ζ⁷

+ 67

63360h⁷ζ⁸− 1

3540h⁸ζ⁹]f_n+1+ [− 91

12hζ²+ 1361

120h²ζ³− 12325 1728h³ζ⁴ +13159

5400 h⁴ζ⁵− 635

1296h⁵ζ⁶+ 33

70h + 439

7560h⁶ζ⁷− 13

3456h⁷ζ⁸+ 1

9720h⁸ζ⁹]f_n+2 + [65

6hζ²−13177

756h²ζ³+ 143

12h³ζ⁴− 1051 240h

4

ζ⁵+ 269

288h⁵ζ⁶− 86 35h

− 13

112h⁶ζ⁷+ 1

128h⁷ζ⁸− 1

4536h⁸ζ⁹]f_n+3+ [91

8hζ²+ 284

15h²ζ³− 38989 2880h³ζ⁴ + 9409

1800h

4

ζ⁵− 505

432h⁵ζ⁶+ 369

700h + 191

1260h⁶ζ⁷− 61

5760h⁷ζ⁸+ 1

3240h⁸ζ⁹]fn+4

+ [ 91

10hζ²− 309

30h²ζ³+ 4087

360h³ζ⁴− 16313 3600h

4

ζ⁵+ 454

4320h⁵ζ⁶− 396 175h

− 713

5040h⁶ζ⁷+ 99

5760h⁷ζ⁸− 1

3240h⁸ζ⁹]f_n+5+ [ 91

12hζ²+ 14087 1080h²ζ³

− 9371

960h³ζ⁴+ 2393 600h

4

ζ⁵− 137

144h⁵ζ⁶− 299

700h + 37

280h⁶ζ⁷− 19 1920h⁷ζ⁸

+ 1

3340h⁸ζ⁹]f_n+6+ [2048

429hζ²− 11264

1365h²ζ³+ 120064

19305h³ζ⁴− 247552 96525h

4

ζ⁵

+ 7168

11583h⁵ζ⁶− 12288

25025h − 11776

135135h⁶ζ⁷+ 128

19305h⁷ζ⁸− 256

1216215h⁸ζ⁹]f_n+¹³

2

+ [− 13

14hζ²+ 677

420h²ζ³− 11

9h³ζ⁴+ 1829 3600h

4

ζ⁵− 107

864h⁵ζ⁶+ 89 5040h

− 11

8064h⁶ζ⁷+ 1

22680h⁷ζ⁸− 1

4536h⁸ζ⁹]f_n+7. (16)

Evaluating (16) atζ = 0, h,2h,3h,4h,5h, ¹³₂ and 7hyields the coefficients of the discrete HOBIM method k=7 in Table2.

490J.P.Chollom,G.M.Kumleng,S.Longwap

Table 2: Coefficients of HOBIM k= 7

k 1 2 3 4 5 6 ¹³₂ 7

α₀ 0 0 0 0 0 1 0 0

α₆ 1 1 1 1 1 0 1 1

β_{0 31135104}²⁴¹⁷²⁵ ₄₆₇₇₇₅⁵²⁸ _3203200²¹⁶⁷ _24324300¹³⁵³ _778377600³³²²⁷⁷ ₁₀₀₁₀₀²⁸⁵²³ 19926466560^19090599 115467 778377600

β₁ -_31135104^1111685 -₄₆₇₇₇₅⁸³²⁸ -_3203200²⁴⁶⁸⁷ -_24324300¹⁹⁹⁶ -_778377600^{332277 160992}₁₀₀₁₀₀ -19926466560^187982769 -_778377600^1109667 β₂ -_31135104^{4075875 185306}_{467775 3203200}¹⁵⁰⁵⁷⁹ _24324300^8148291 _778377600^15870569 -₁₀₀₁₀₀⁴⁷¹⁹ 19926466560^851282003 4849559

778377600

β₃ -^21503625_31135104 ⁵⁷¹⁵⁶⁰₄₆₇₇₇₅ -^1522235_3203200 -_24324300⁸⁴⁰⁸⁴⁰ -_778377600^{47775585 245960}₁₀₀₁₀₀ -19926466560^2384941845 -_778377600^12833535 β₄ -^39006825_31135104 ³⁷⁶⁷²⁸₄₆₇₇₇₅ -_3203200^{343500 10470603}_24324300 ^112965567_778377600 -₁₀₀₁₀₀⁵²⁷⁶⁷ 19926466560^4822595349 23326017

778377600

β₅ -^27144975_31135104 ^5578832₄₆₇₇₇₅ -^3308877_3203200 ^28918032_24324300 -^476877687_778377600 ²²⁶⁵¹²₁₀₀₁₀₀ -19926466560^8698280019 -_778377600^31923177 β₆ -^22250085_31135104 ²¹⁷³³⁸₄₆₇₇₇₅ -^1853423_3203200 ^11821953_24324300 -^482889693_778377600 -₁₀₀₁₀₀⁴²⁷⁵⁷ 19926466560^6050421649 -_778377600^84241443 β¹³

2

6963200

31135104 -₄₆₇₇₇₅³²⁷⁶⁸ _3203200⁴⁴²³⁶⁸ -_24324300^{2080768 119554048}_{778377600 100100}⁴⁹¹⁵² 46852243456

19926466560 -^556285952_778377600 β₇ -_31135104^1104675 ₄₆₇₇₇₅²⁹⁰⁴ -_3203200⁶⁰⁴⁸⁹ _24324300²²⁹⁹⁴⁴ -_778377600^16195179 -₁₀₀₁₀₀¹⁰²⁹⁶ -19926466560^2146195623 -^120274869_778377600

2.3. Derivation of HOBIM k= 8

Consideringk= 8,x ∈[xn, x_n+8] The continuous form of the method HOBIM k=8 is given by

y(x) =

k

X

j=0

ψ_j(x)y_n+j+h

k

X

j=0

β_j(x)f_n+j+β_j(µ), µ= 15

2 . (17) Evaluating the matrix (8) withs= 9, t= 2 produces the matrix of the method Din (18)

D=















1 x_n+7 x²_n+7 . . . x¹⁰_n+7 0 1 2x_n . . . 10x⁸_n

... ... ... ... ... 0 1 2xn+µ . . . 9x⁸_n+µ 0 1 2x_n+7 . . . 10x⁹_n+7















(18)

Inverting (18) using the Maple software yields the elements of the matrix C .Substituting the elements of of C into (17) produces the continuous form of the multistep collocation method for k=8.which are substituted into (14) to give the continuous form of the multistep collocation method. Evaluating the method atζ = 0, h,2h,3h,4h,5h,6h¹⁵₂ and8h yields the coefficients of the discrete HOBIM method k=8 in Table3.

2.4. Derivation of HOBIM k= 9

Consideringk= 9,x ∈[x_n, x_n+9] The continuous form of the method HOBIM k=9 is given by

y(x) =

k

X

j=0

ψ_j(x)y_n+j+h

k

X

j=0

β_j(x)f_n+j+β_j(µ), µ= 17

2 (19)

Evaluating the matrix (8) withs= 10, t= 2 produces the matrix of the method Din (20)

D=















1 x_n+8 x²_n+8 . . . x¹¹_n+8 0 1 2x_n . . . 11x¹⁰_n

... ... ... ... ... 0 1 2x_n+µ . . . 11x¹⁰_n+µ 0 1 2x_n+9 . . . 11x¹⁰_n+9















(20)

492J.P.Chollom,G.M.Kumleng,S.Longwap

Table 3: Coefficients of HOBIM k=8

k 1 2 3 4 5 6 7 ¹⁵₂ 8

α₀ 0 0 0 0 0 0 1 0 0

α₇ 1 1 1 1 1 1 0 1 1

β_{0 140}⁹ -_10878368¹²¹⁵⁵ ₇₇₉₆₂₅¹⁴³ -_1601600⁴⁷¹⁹ _66237000¹²¹ -_259459200⁶⁵⁹²³ _18528000⁷³⁸⁵⁸⁰ -10218700800^6636839 148577 1297296000

β₁ -⁴⁸²₁₄₀ -_10878368¹⁹²⁷⁷⁵ -₇₇₉₆₂₅²²⁰⁰ -_1601600⁵⁴³⁹⁵ -_6237000⁷⁰ _259459200⁷¹⁹²⁷⁹ _185328000^44021285 -10218700800^71053180 -_1297296000^1563925 β₂ -¹⁹⁰⁸₁₄₀ -_10878368^4167995 ₇₇₉₆₂₅¹⁹³⁴⁰ -_1601600³⁰³⁶¹⁵ -_6237000⁵⁰ -_259459200^3637699 -_185328000⁷⁰⁸⁶²⁵ 10218700800^349808000 7522385

1297296000

β₃ -⁷⁷⁴₁₄₀ -^12451725_10878368 -³²²¹⁹⁰₇₇₉₆₂₅ -^1193335_{1601600 6237000}²⁶⁷³⁰ _259459200^11443008 _185328000^78863785 -10218700800^1055993180 -_1297296000^21949785 β₄ -²⁰⁹⁰₁₄₀ -_10878368^8991125 -⁹³¹¹⁵⁰₇₇₉₆₂₅ -^8319025_1601600 -_6237000¹⁹⁷⁴⁵ -_259459200^25813645 -_185328000^34809775 10218700800^2218572950 43675775

1297296000

β₅ -⁷⁷⁴₁₄₀ -^10960235_10878368 -⁶⁵¹⁹⁰⁴₇₇₉₆₂₅ -^16382223_1601600 ^2664442_{6237000 259459200}^48730253 _185328000^79718639 -10218700800^3588114772 -_1297296000^63837631 β₆ -¹⁹⁰⁸₁₄₀ -^10750025_10878368 -⁹⁰⁹⁷⁰⁰₇₇₉₆₂₅ -^17205045_1601600 ^7431820_6237000 -^198188449_259459200 -_185328000⁸⁴⁴³⁴³ 10218700800^5389813880 74006075

1297296000

β₇ -⁴⁸²₁₄₀ -_10878368⁵⁹⁸⁰⁹⁵ -³⁷⁹³⁹⁰₇₇₉₆₂₅ -^8700835_1601600 ^3016750_6237000 -^1530552471_{259459200 185328000}^41487875 -31824380500

10218700800 122573165 1297296000

β15

2 0 _10878368^1392640 ₇₇₉₆₂₅^{65536 1851392}_1601600 -_6237000⁵²⁴²⁸⁸ _259459200^34816000 -_185328000^19152896 -23519854592

10218700800 988491904 1297296000

β₈ -₁₄₀⁹ 0 -₇₇₉₆₂₅⁶⁹⁸⁵ -_1601600²³¹⁶⁶⁰ _6237000⁵⁷¹⁴⁵ -_259459200^4409548 _185328000^3343340 10218700800^1001060555 198229460 1297296000

Inverting (20) using the Maple software yields the elements of the matrix C.Substituting the elements of the continuous coefficients into (19) produces the continuous form of the multistep collocation method for k = 9. Evalu- ating the continuous form of the linear multi-step method at ζ =x−x_n, ζ = 0, h,2h,3h,4h,5h,6h,7h,¹⁷₂ and 9hproduces the coefficients of the discrete HO- BIM method for k= 9 inT able4 used as block integrators.

2.5. Derivation of HOBIM k = 10

Consideringk= 10,x∈[xn, x_n+10] The continuous form of the method HOBIM k= 10 is given by

y(x) =

k

X

j=0

ψ_j(x)y_n+j+h

k

X

j=0

β_j(x)f_n+j+β_j(µ), µ= 19

2 (21)

Evaluating the matrix (8) withs= 10, t= 2 produces the matrix of the method Din (22)

D=















1 x_n+9 x²_n+9 . . . x¹²_n+9 0 1 2xn . . . 12x¹¹_n

... ... ... ... ... 0 1 2x_n+µ . . . 12x¹¹_n+µ 0 1 2xn+9 . . . 12x¹¹_n+9















(22)

Inverting [22] using the Maple software yields the elements of the matrix (9).Sub- stituting the elements of the continuous corfficients into(21) produces the con- tinuous form of the multistep collocation method fork= 10.Evaluating the con- tinuous form of the linear multi-step method at ζ =x−x_n, ζ = 0, h,2h,3h,4h, 5h,6h,7h,8h,¹⁹₂ and 10hproduces the coefficients of the discrete HOBIM method fork= 10 in Table 5 used as block integrators.

3. Analysis of the New Methods

In this section, we determine the convergence, construct the regions of absolute stability and obtain the orders of the new HOBIM methods.

3.1. Convergence Analysis

Using the approach of Fatunla [15],[16] we determine the convergence of the new HOBIM methods where the block methods are represented as a single block, r

494J.P.Chollom,G.M.Kumleng,S.Longwap

Table 4: Coefficients of HOBIM k= 9

k 1 2 3 4 5 6 7 8 ¹⁷2 9

α0 0 0 0 0 0 0 0 1 0 0

α8 1 1 1 1 1 1 0 1 1 1

β0

487806319

83175206400-_37437400²⁷¹⁷⁰ 23289057792⁷⁴⁰⁸⁸² -_1137161025⁶⁷²¹ _2395993600³⁷⁴⁸⁰³ -_9097288200³²²³²² 582226444800^94307785

310589708 1137161025

13779794171 2980999939737600

52865527 582226444800

β1-28178875725 83175206400

517808

37437400-23289057792^102794835-_1137161025²⁷⁷¹³⁴ -_2395993600^4528953 _9097288200^3464175 -582226444800^1109870619

1958491392

1137161025- 159914887275 2980999939737600

605909733 582226444800

β2-11533848400

83175206400-^14168242_37437400-23289057792^781136400

3525324 1137161025

25782064

2395993600-_9097288200^16297050 582226444800^6053782416 -_1137161025^616615296 855986646900

2980999939737600-582226444800^319063588

β3-41062788312

83175206400-^47494668_37437400-10314131880

23289057792-_1137161025^29081832-_2395993600^94077048 _9097288200^41113956 -20405861736 582226444800

4247273472

1137161025- 2809327722708 2980999939737600

810238952984 582226444800

β4-131660563034

83175206400-^27163994_37437400-26179560550 23289057792

471545932 1137161025

267726030

2395993600-_9097288200^37724258 48123351562

582226444800-^3068699920_1137161025 6360134317394

2980999939737600-22433058934 582226444800

β5-39910344474

83175206400-^47574670_37437400-22441654950 23289057792

1356118764

1137161025-^1359215858_2395993600-_9097288200^189370038-86784001590 582226444800

5440461312

1137161025- 10688638026066 2980999939737600

35778539082 582226444800

β6-111126591576

83175206400-^28457286_37437400-22480234920 23289057792

952786692

1137161025-^2343736824_2395993600^3794299938_9097288200 136284991128

582226444800-^2232008064_1137161025 14402757628116

2980999939737600-43748723304 582226444800

β7-72986593680

83175206400-^45479148_37437400-25633436400 23289057792

1325517336

1137161025-^2650334544_239599360010905806340

9097288200-396653667696 582226444800

3270998016

1137161025- 18534243061620 2980999939737600

43998493968 582226444800

β8-55703941293

83175206400-^17046172_37437400-12237179955 23289057792

55440656

1137161025-_2395993600^124300067 _9097288200^435091284-329020653819

582226444800-_1137161025^718827252 94946391302763 2980999939737600

46922410821 582226444800

β¹⁷

2

154559024896 83175206400

2490368 37437400

2446458880

23289057792-_1137161025^96206848 _2395993600^242614272 -_9097288200^735182848 69482971136 582226444800

620756992 1137161025

6734908122764 2980999939737600

426040754176 582226444800

β⁹-83175206400^2210729521-37437400²²¹²²¹-23289057792^291294575

10302578

1137161025-2395993600^28284685

77872223

9097288200-582226444800^8291793367-^11551321601137161025- 2686037350139 2980999939737600

88066667689 582226444800

ORDERBLOCKIMPLICITMULTI-STEP(HOBIM)...495 Table 5: Coefficients of HOBIM k= 10

k 1 2 3 4 5 6 7 8 9 ¹⁹2 10

α0 0 0 0 0 0 0 0 0 1 0 0

α9 1 1 1 1 1 1 1 1 1 1 1

β0 2368 467775

104261612 143666266600

663 6806800

5220020 40226554368

58123 1964187225

391612

4138534400-1654052400⁵⁸¹²³ -1005663859200^109454228

1110235932

4138534400-242305833369600^8279227567

74000524 100663859200

β1-¹⁵⁴²²⁸₄₆₇₇₇₅-143666266600^2007851391-_6806800¹²³⁵⁰-40226554368^71129825-_1964187225⁶³⁸⁰⁰¹-_4138534400^5024227 _1654052400⁷⁰⁸²¹⁴ 1005663859200^1386207225

7351165341 4138534400

108733231706

242305833369600-100663859200^918377863

β2-⁶⁷⁰⁶⁵⁶467775

5461551540 143666266600

13193 6806800

475095855 40226554368

2834325 1964187225

3014462

4138534400-1654052400^3942861-1005663859200^8155940247 -^33756810754138534400- 601900264719 242305833369600

5264601225 100663859200

β3-¹⁶¹⁶⁶⁴₄₆₇₇₇₅180925400880

143666266600-^2692324_6806800-40226554368^2264746800-_1964187225^4104684-_4138534400^113215376_1654052400^13156980 29653120944 1005663859200

18904197744 4138534400

2147824384920

242305833369600-18489651792 100663859200

β4-⁸⁸⁸¹⁹²467775

10874516452 143666266600-^83949066806800

19708762320

40226554368-1964187225^29149458

30454507

4138534400-1654052400^28558062- 74948522064

1005663859200-18192285072

4138534400- 5286619356798 242305833369600

44527472496 100663859200

β5 254 467775

171946522202

143666266600-^5291624_680680042442190310 40226554368

78356699

1964187225-_4138534400^670773454_1654052400^37780392 141351801994 1005663859200

29860860978 4138534400

9574283911868

242305833369600-78116289654 100663859200

β6-⁸⁸⁸¹⁹²467775

128365168386 143666266600-^82720306806800

4173826995 40226554368

237551706 1964187225

2570955322 4138534400

1300806

1654052400-212288549070

1005663859200-18436944006

4138534400-13407250515126 242305833369600

103979593458 100663859200

β7-¹⁶¹⁶⁶⁴467775

144402635832 143666266600-^54763806806800

36449167560 40226554368

161921838 1964187225

3869697624

4138534400-1654052400^663368628

285312476280 1005663859200

19277944920 4138534400

15482382099288

242305833369600-109310215560 100663859200

β8-⁶⁷⁰⁶⁵⁶467775

155782312140 143666266600-^80799816806800

45763641300 40226554368

2306094999 1964187225

4689459996

4138534400-^19994389351654052400-716323517364

1005663859200-^37535532844138534400-17424846944115 242305833369600

97087548012 100663859200

β9 154228 467775

76406192317

143666266600-^3225274_680680040226554368^2014514339

946567973 1964187225

2074839273

4138534400-_1654052400^780032318-547512098731 1005663859200

7071716457 4138534400

78748729590778 242305833369600

66987700949 100663859200

β¹⁹

2 0 -15459024896 143666266600

524288

6806800-40226554368^3662807040-1964187225^159907840-4138534400^376832000

127401984 1654052400

108213174272

1005663859200-^30611865604138534400

538550857537536 242305833369600

743868334080 100663859200

β¹⁰ 0 143666266600^183756218-6806800⁵²⁸¹⁹

403964795 40226554368

1669102 1964187225

41414737

4138534400-1654052400^13054249- 12242558211 1005663859200

490338225

4138534400- 2025997762971 242305833369600

1507122610825 100663859200