- Research
- Open Access
G-stability one-leg hybrid methods for solving DAEs
- P. Agarwal^{1, 2, 3}Email authorView ORCID ID profile,
- Iman H. Ibrahim^{4} and
- Fatma M. Yousry^{4}
https://doi.org/10.1186/s13662-019-2019-2
© The Author(s) 2019
- Received: 20 September 2018
- Accepted: 12 February 2019
- Published: 12 March 2019
Abstract
This paper introduces the solution of differential algebraic equations using two hybrid classes and their twin one-leg with improved stability properties. Physical systems of interest in control theory are sometimes described by systems of differential algebraic equations (DAEs) and ordinary differential equations (ODEs) which are zero index DAEs. The study of the first hybrid class includes the order of convergence, A(α)-stability, stability regions, and G-stability for its one-leg twin in two cases: for step (\(k=1\)) and steps (\(k=2\)). For the second class, G-stability of its one-leg twin is studied in two cases: for steps (\(k=2\)) and steps (\(k=3\)). Test problems are introduced with different step size at different end points.
Keywords
- Hybrid methods
- One-leg methods
- DAEs
- G-stability
- A(α)-stability
MSC
- 65L05
- 65L06
- 65L20
1 Introduction
A fundamentally important concept in the algorithms of the numerical solutions of DAEs is the index of a DAE. In a sense, this tells us how far away the DAE is from being an ODE. The index of a DAE is the minimum number of times all or part of the DAE system must be differentiated with respect to time in order to convert the DAE into an explicit ODE. The higher the index is, the further it is from an ODE and the more difficult it is in general to solve the DAE [12].
The first general method applied to the numerical solution of DAEs is backward differentiation formula (BDF). Ebadi andGokhale presented in [9–11] class \(2+1\) hybrid BDF-like methods, hybrid BDF methods (HBDF), and new hybrid methods for the numerical solution of IVPs. These methods have wide stability regions and good performance in solving CPU time compared to the extended BDF (EBDF) and modified extended BDF (MEBDF) methods [3].
In Sect. 2 the first hybrid class is derived, its orders of convergence are investigated, and its stability analysis is studied. In Sect. 3 some basic notions of one-leg schemes and G-stability are mentioned. The one-leg twin of the first class is derived and its G-stability is discussed in Sect. 4. The one-leg twin of the second class [14] is derived and its G-stability is discussed in Sect. 5. Numerical tests are investigated in Sect. 6. Finally, a conclusion is introduced.
2 The first hybrid class
The coefficients of method (2.2) for orders 2, 3, and 4
k | 1 | 2 | 3 |
---|---|---|---|
\(\alpha _{n - 1}\) | −1 | \(\frac{2( - 8 - 6s+ 3\beta _{0} + 3s\beta _{0})}{14 + 9s}\) | \(\frac{ - 18(12 + s(15 + 4s)) + (1 + s)(119 + 46s)\beta _{0}}{2(85 + 90s+ 22s)} \) |
\(\alpha _{n - 2}\) | 0 | \(\frac{2 + 3s- 6(1 + s)\beta _{0}}{14 + 9s}\) | \(\frac{9(3 + 2s(3 + s)) - 4(1 + s)(17 + 7s)\beta _{0}}{85 + 90s+ 22s}\) |
\(\alpha _{n - 3}\) | 0 | 0 | \(\frac{ - 2(4 + s(9 + 4s)) + (1 + s)(17 + 10s) \beta _{0}}{2(85 + 90s+ 22s)} \) |
\(\beta _{1}\) | \(\frac{1 + 2s - 2(1 + s)\beta _{0}}{2s}\) | \(\frac{4 + 6s(2 + s) - (1 + s)(5 + 3s)}{s(14 + 9s)}\) | \(\frac{ - 6(3 + 2s)(1 + s(3 + s)) + (1 + s)(17 + s(17 + 4s))\beta _{0}}{s(85 + 90s+ 22s)} \) |
\(\beta _{s}\) | \(- \frac{1 - 2\beta _{0}}{2s}\) | \(\frac{ - 4 + 5\beta _{0}}{s(14 + 9s)}\) | \(\frac{ - 18 + 17\beta _{0}}{s(85 + 90s+ 22s)} \) |
The coefficients of method (2.2) for order 5
k | 4 |
---|---|
\(\alpha _{n - 1}\) | \(\frac{ - 288(48 + s(78 + s(36 + 5s))) + (1 + s)(8996 + 6055s+ 985s)\beta _{0}}{6(1660 + s(2265 + s(952 + 125s)))} \) |
\(\alpha _{n - 2}\) | \(\frac{72(24 + s(57 + s(32 + 5s))) - 3(1 + s)(1274 + s(913 + 155s)) \beta _{0}}{(3320 + 2s(2265 + s(952 + 125s)))} \) |
\(\alpha _{n - 3}\) | \(\frac{ - 32(8 + 5s)(2 + s(4 + s)) + 3(1 + s)(316 + s(281 + 55s)) \beta _{0}}{(3320 + 2s(2265 + s(952 + 125s)))} \) |
\(\alpha _{n - 4}\) | \(\frac{18(12 + s(33 + s(24 + 5s))) - (1 + s)(374 + s(367 + 85s))\beta _{0}}{6(1660 + s(2265 + s(952 + 125s)))} \) |
\(\beta _{1}\) | \(\frac{12(24 + 5s(4 + s)(5 + s(4 + s))) - 3(1 + s)(74 + s(96 + s(39 + 5s)))\beta _{0}}{s(1660 + s(2265 + s(952 + 125s)))} \) |
\(\beta _{s}\) | \(\frac{6( - 48 + 37\beta _{0})}{s(1660 + s(2265 + s(952 + 125s)))} \) |
The coefficients of method (2.2) for order 6
k | 5 |
---|---|
\(\alpha _{n - 1}\) | \(\frac{( - 7200(120 + s(231 + s(142 + s(35 + 3s)))) + (1 + s)(615{,}436 + s(553{,}907 + 3s(53{,}483 + 5002s)))\beta _{0})}{(12(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s))))}\) |
\(\alpha _{n - 2}\) | \(\frac{ (300(120 + s(321 + s(236 + s(65 + 6s)))) - 4(1 + s)(17{,}929 + s(17{,}174 + s(5194 + 501s)))\beta _{0})}{(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s)))} \) |
\(\alpha _{n - 3}\) | \(\frac{ ( - 400(40 + s(117 + s(98 + 3s(10 + s)))) + 9(1 + s)(2948 + s(3325 + s(1127 + 118s)))\beta _{0})}{(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s)))} \) |
\(\alpha _{n - 4}\) | \(\frac{(225(60 + s(183 + s(164 + s(55 + 6s)))) - 4(1 + s)(5221 + s(6362 + 3s(794 + 91s)))\beta _{0}}{ (3(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s))))}\) |
\(\alpha _{n - 5}\) | \(\frac{( - 96(24 + s(75 + s(70 + s(25 + 3s)))) + (1 + s)(3436 + s(4379 + s(1753 + 222s))))\beta _{0}}{(4 (48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s))))}\) |
\(\beta _{1}\) | \(\frac{(60(5 + 2s)(24 + s(5 + s)(20 + 3s(5 + s))) - 24(1 + s)(197 + s(302 + s(163 + s(37 + 3s))))\beta _{0})}{(s(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s))))}\) |
\(\beta _{s}\) | \(\frac{ - 7200 + 4728\beta _{0}}{ s(48{,}076 + s(77{,}175 + s(42{,}980 + 9975s+ 822s)))}\) |
The coefficients of method (2.2) for order 7
k | 6 |
---|---|
\(\alpha _{n - 1}\) | \(\frac{ - 7200(1440 + s(3132 + s(2320 + s(775 + s(120 + 7s))))) + (1 + s)(7{,}796{,}104 + s(8{,}479{,}052 + s(3{,}336{,}768 + 7s(80{,}701 + 4973s))))\beta _{0})}{(60(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))}\) |
\(\alpha _{n - 2}\) | \(\frac{ (300(720 + s(2106 + s(1844 + s(685 + s(114 + 7s))))) - 5(1 + s)(78{,}796 + s(91{,}175 + s(37{,}473 + s(6547 + 413s))))\beta _{0})}{(215{,}824 + 2s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \) |
\(\alpha _{n - 3}\) | \(\frac{ ( - 1200(480 + s(1524 + s(1488 + s(605 + s(108 + 7s))))) + 5(1 + s)(174{,}152 + s(230{,}788 + s(104{,}616 + 7s(2807 + 187s))))\beta _{0})}{(9(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))} \) |
\(\alpha _{n - 4}\) | \(\frac{225(360 + s(1188 + s(1228 + s(535 + s(102 + 7s))))) - 20(1 + s)(5701 + s(8066 + s(3942 + s(793 + 56s))))\beta _{0}}{3(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \) |
\(\alpha _{n - 5}\) | \(\frac{ - 96(288 + s(972 + s(1040 + s(475 + s(96 + 7s))))) + 5(1 + s)(7496 + s(11{,}020 + s(5664 + 7s(173 + 13s))))\beta _{0}}{ 4(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))}\) |
\(\alpha _{n - 6}\) | \(\frac{300(240 + s(822 + s(900 + s(425 + s(90 + 7s))))) - (1 + s)(95{,}356 + s(143{,}753 + s(76{,}527 + s(17{,}173 + 1379s))))\beta _{0}}{ 90(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))}\) |
\(\beta _{1}\) | \(\frac{(60(720 + 7s(6 + s)(84 + s(6 + s)(17 + s(6 + s)))) - 10(1 + s)(2484 + s(4292 + s(2785 + s(855 + s(125 + 7s)))))\beta _{0})}{ (3s(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s))))))}\) |
\(\beta _{s}\) | \(\frac{360( - 40 + 23 \beta _{0})}{s(107{,}912 + s(194{,}628 + s(130{,}060 + s(40{,}775 + s(6054 + 343s)))))} \) |
Since formula (3) is of order k and formula (4) is of order \(k+1\), then it is easy to see that method (3)–(4) has order \(k+1\).
2.1 Stability analysis
A(α)-stability for BDF, EBDF, A-EBDF, MEBDF, Enright methods, HEBDF, and The class (3–4) for various orders
k | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
BDF | |||||||
Order | 1 | 2 | 3 | 4 | 5 | 6 | - |
α | 90° | 90° | 88° | 73° | 51° | 18° | - |
EBDF | |||||||
Order | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
α | 90° | 90° | 90° | 87.61° | 80.2° | 67.7° | 48.8° |
A-EBDF | |||||||
Order | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
α | 90° | 90° | 90° | 88.85° | 84.2° | 75° | 61° |
MEBDF | |||||||
Order | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
α | 90° | 90° | 90° | 88.4° | 82.5° | 74.5° | 62° |
Enright methods | |||||||
Order | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
α | 90° | 90° | 87.88° | 82.03° | 73.10° | 59.95° | 37.61° |
HEBDF | |||||||
Order | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
α | 90° | 90° | 90° | 89.013° | 85.2° | 77.195° | 60.686° |
The class (3–4) | |||||||
Order | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
α | 90° | 90° | 90° | 90° | 86.1^{o} | 81.6^{o} | 75.2^{o} |
We recall some basic notions of one-leg schemes and G-stability.
3 One-leg schemes
3.1 G-stability analysis
The G-stability analysis, announced at the 1975 Dundee conference and published in [6], uses the test problem \(\mathrm{d}y/\mathrm{d}x = f(x, y)\), where \(\langle y - z, f(x, y) - f(x, z)\rangle \leq 0\). In the same publication, one-leg methods were introduced and related to corresponding linear multistep methods. Stable behavior for this problem was defined as G-stability. A more detailed account of this work will be given.
Definition 1
([6])
Theorem 2
([12])
Theorem 3
([5])
If ρ and σ have no common divisor, then the method \(( \rho , \sigma )\) is A-stable if and only if the corresponding one-leg method is G-stable.
4 One-leg method for the first hybrid class
Here, the one-leg twin of the first class is studied when \(k=1\) and \(k=2\).
5 The second hybrid class
Here, the one-leg twin of the second class is studied when \(k = 2\) and \(k = 3\).
6 Numerical tests
Here, some numerical results are presented to evaluate the performance of the proposed technique [1, 4, 13]. The numerical tests are solved after reduction.
Test 1
Test 2
Test 3
The error of the first test
t | h | \(\operatorname{Er}(y_{1}(t))\) | \(\operatorname{Er}(y_{2}(t))\) | |
---|---|---|---|---|
Hybrid Method (4) | 3 | 0.001 | 4.78965E-8 | 1.53733E-8 |
6 | 0.001 | 2.07019E-6 | 3.41682E-7 | |
3 | 0.0001 | 1.67133E-10 | 4.69775E-11 | |
6 | 0.0001 | 1.26784E-9 | 1.45462E-10 | |
One-Leg of Hybrid Method (4) | 3 | 0.001 | 8.87175E-6 | 4.70979E-6 |
6 | 0.001 | 4.45514E-4 | 8.88051E-5 | |
3 | 0.0001 | 8.92249E-8 | 4.72626E-8 | |
6 | 0.0001 | 4.48537E-6 | 8.9293E-7 | |
Hybrid Method (27) | 3 | 0.001 | 1.15024E-7 | 3.66907E-8 |
6 | 0.001 | 5.31317E-6 | 8.72459E-7 | |
3 | 0.0001 | 3.5984E-10 | 1.107355E-10 | |
6 | 0.0001 | 1.12391E-8 | 1.93154E-9 | |
One-Leg of Hybrid Method (27) | 3 | 0.001 | 1.28838E-6 | 9.03448E-8 |
6 | 0.001 | 2.06502E-4 | 2.96481E-5 | |
3 | 0.0001 | 1.22774E-8 | 1.09555E-9 | |
6 | 0.0001 | 2.02582E-6 | 2.90166E-7 | |
HBDF | 3 | 0.001 | 4.67957E-9 | 1.52812E-8 |
6 | 0.001 | 7.17848E-6 | 1.01547E-6 | |
3 | 0.0001 | 2.87855E-10 | 9.16494E-11 | |
6 | 0.0001 | 4.06857E-8 | 6.55177E-9 |
The error of the second test
t | h | \(\operatorname{Er}(y_{1}(t))\) | \(\operatorname{Er}(y_{2}(t))\) | |
---|---|---|---|---|
Hybrid Method | 1.3 | 0.001 | 6.79234E-13 | 4.38633E-10 |
2.3 | 0.001 | 3.66441E-8 | 1.53328E-8 | |
1.5 | 0.0001 | 1.23906E-12 | 7.22089E-13 | |
3 | 0.0001 | 7.66911E-10 | 1.98713E-9 | |
One-Leg Hybrid Method | 1.3 | 0.001 | 1.20205E-6 | 1.07885E-6 |
2.3 | 0.001 | 1.08490E-5 | 3.18015E-6 | |
1.5 | 0.0001 | 1.54044E-8 | 1.13384E-8 | |
3 | 0.0001 | 8.71812E-8 | 2.76648E-8 | |
Class1 | 1.3 | 0.001 | 8.48159E-11 | 4.33369E-10 |
2.3 | 0.001 | 1.37322E-7 | 3.42503E-8 | |
1.5 | 0.0001 | 1.41808E-12 | 2.87503E-12 | |
3 | 0.0001 | 1.36406E-8 | 3.2887E-8 | |
One-Leg Class1 | 1.3 | 0.001 | 2.17487E-7 | 1.48626E-7 |
2.3 | 0.001 | 2.25127E-6 | 1.99805E-6 | |
1.5 | 0.0001 | 3.73525E-9 | 6.91022E-10 | |
3 | 0.0001 | 2.8067E-7 | 1.95566E-6 | |
HBDF | 1.3 | 0.001 | 3.24911E-9 | 2.8293E-9 |
2.3 | 0.001 | 8.05334E-6 | 1.9571E-6 | |
1.5 | 0.0001 | 4.8609E-11 | 3.10769E-11 | |
3 | 0.0001 | 7.08909E-3 | 2.92056E-1 |
The error of the third test
t | h | \(\operatorname{Er}(x_{1}(t))\) | \(\operatorname{Er}(x_{2}(t))\) | Er(y(t)) | |
---|---|---|---|---|---|
Hybrid Method | 1 | 0.001 | 1.22665E-10 | 1.44872E-10 | 2.37014E-9 |
3 | 0.001 | 8.34595E-8 | 2.85294E-7 | 8.03989E-7 | |
1 | 0.0001 | 1.63203E-13 | 2.12274E-13 | 2.45715E-12 | |
3 | 0.0001 | 1.02396E-10 | 3.50464E-10 | 9.74921E-10 | |
One-Leg Hybrid Method | 1 | 0.001 | 3.66007E-8 | 9.76405E-8 | 6.03067E-7 |
3 | 0.001 | 1.38646E-7 | 2.52238E-7 | 3.53319E-6 | |
1 | 0.0001 | 3.62941E-10 | 9.81447E-10 | 5.9809E-9 | |
3 | 0.0001 | 3.0906E-10 | 1.16064E-9 | 2.49718E-8 | |
Class1 | 1 | 0.001 | 5.5670E-6 | 6.86347E-6 | 3.45989E-5 |
3 | 0.001 | 3.41734E-5 | 1.18007E-4 | 2.28748E-4 | |
1 | 0.0001 | 8.08261E-8 | 9.76428E-8 | 4.405E-7 | |
3 | 0.0001 | 3.46347E-7 | 1.19601E-6 | 2.32459E-6 | |
One-Leg Class1 | 1 | 0.001 | 6.7857E-9 | 1.47176E-8 | 1.12592E-7 |
3 | 0.001 | 1.69542E-7 | 5.4232E-7 | 1.85398E-6 | |
1 | 0.0001 | 2.48221E-7 | 1.85705E-7 | 7.07125E-8 | |
3 | 0.0001 | 3.66093E-9 | 1.22613E-8 | 3.38426E-8 | |
HBDF | 1 | 0.001 | 5.68471E-10 | 1.28144E-9 | 6.63956E-9 |
3 | 0.001 | 1.07333E-8 | 4.18462E-8 | 8.94225E-8 | |
1 | 0.0001 | 7.12319E-13 | 1.52255E-12 | 6.35358E-12 | |
3 | 0.0001 | 1.43312E-11 | 5.25517E-11 | 1.13875E-10 |
7 Conclusion
In this paper, the first hybrid class is studied for \(k=1\) up to 6. It has large stability regions. Its one-leg twin for \(k=1\) (\(p=2\)) and \(k=2\) (\(p=3\)) is G-stable. In the second class, for \(k = p = 2\), the one-leg twin has order 2 except when \(s =\frac{1}{3} (-3+\sqrt{3} \sqrt{(1 -2 \beta ^{*} + \beta ^{*2})})\) it has order 3. For \(k = p = 3\), the one-leg twin has order 2 and if \(\beta^{*} = 0\), it leads to one leg hybrid BDF, and they are G-stable. The numerical tests show that the hybrid method (4) gives good result with small steps.
Declarations
Acknowledgements
We are really thankful to the reviewers for their useful suggestions and corrections.
Availability of data and materials
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Funding
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Authors’ contributions
All authors contributions are equal and they all read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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