Journal of Mathematical Chemistry

, Volume 56, Issue 10, pp 3014–3044

# New Runge–Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems

• Ruru Hao
• T. E. Simos
Original Paper

## Abstract

A new three-stages symmetric two-step method with improved properties is developed in this paper and for the first time in the literature. The properties of the new proposed algorithm are:
• is a symmetric finite difference pair,

• is a scheme of two-step,

• is an algorithm of three-stages—i.e. hybrid or Runge–Kutta type,

• is of tenth-algebraic order,

• it has vanished the phase-lag and its first, second and third derivatives,

• it has improved stability properties for the general problems,

• it is a P-stable method since it has an interval of periodicity equal to $$\left( 0, \infty \right)$$.

The new proposed scheme is constructed based on the following layers:
• An approximation denoted on the first layer on the point $$x_{n-1}$$,

• An approximation denoted on the second layer on the point $$x_{n}$$ and finally,

• An approximation denoted on the third (final) layer on the point $$x_{n+1}$$,

For the new proposed method we give a full theoretical analysis which consists of: (1) local truncation error analysis, (2) comparative local truncation error analysis, (3) stability analysis and (4) interval of periodicity analysis. The efficiency of the new proposed algorithm is tested on the approximate solution of systems of coupled differential equations arising from the Schrödinger equation.

## Keywords

Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schrödinger equation

## References

1. 1.
A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)Google Scholar
2. 2.
C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)Google Scholar
3. 3.
F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)Google Scholar
4. 4.
A.R. Leach, Molecular Modelling—Principles and Applications (Pearson, Essex, 2001)Google Scholar
5. 5.
P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford University Press, Oxford, 2011)Google Scholar
6. 6.
V.N.K.T.E. Simos, V.N. Kovalnogov, I.V. Shevchuk, Perspective of mathematical modeling and research of targeted formation of disperse phase clusters in working media for the next-generation power engineering technologies. AIP Conf. Proc. 1863, 560099 (2017)
7. 7.
V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Y.A. Khakhalev, A.N. Zolotov, Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations. AIP Conf. Proc. 738, 480004 (2016)
8. 8.
V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow. AIP Conf. Proc. 1648, 850033 (2015)
9. 9.
N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika 1, 49–53 (1998)Google Scholar
10. 10.
V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Modeling and development of cooling technology of turbine engine blades. Int. Rev. Mech. Eng. 9(4), 331–335 (2015)Google Scholar
11. 11.
S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar
12. 12.
J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)
13. 13.
G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)
14. 14.
T.E. Simos, P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)
15. 15.
Z. Chen, C. Liu, T.E. Simos, New three-stages symmetric two step method with improved properties for second order initial/boundary value problems. J. Math. Chem. (2018).
16. 16.
V.N. Kovalnogov, R.V. Fedorov, T.E. Simos, New hybrid symmetric two step scheme with optimized characteristics for second order problems. J. Math. Chem. (2018).
17. 17.
Z.A. Anastassi, T.E. Simos, A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math. 236, 3880–3889 (2012)
18. 18.
A.D. Raptis, T.E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT 31, 160–168 (1991)
19. 19.
J.M. Franco, M. Palacios, J. Comput. Appl. Math. 30, 1 (1990)
20. 20.
J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (Wiley, New York, 1991), pp. 104–107Google Scholar
21. 21.
E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)
22. 22.
G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two new optimized eight-step symmetric methods for the efficient solution of the Schrödinger equation and related problems. MATCH Commun. Math. Comput. Chem. 60(3), 773–785 (2008)Google Scholar
23. 23.
G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009)
24. 24.
25. 25.
T.E. Simos, P.S. Williams, Bessel and Neumann fitted methods for the numerical solution of the radial Schrödinger equation. Comput. Chem. 21, 175–179 (1977)
26. 26.
T.E. Simos, Jesus Vigo-Aguiar, A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Commun. 152, 274–294 (2003)
27. 27.
T.E. Simos, G. Psihoyios, J. Comput. Appl. Math. 175(1), IX (2005)
28. 28.
T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math. 19, 65–75 (1972)
29. 29.
R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)
30. 30.
A. Konguetsof, T.E. Simos, A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 93–106 (2003)
31. 31.
Z. Kalogiratou, T. Monovasilis, T.E. Simos, Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 83–92 (2003)
32. 32.
Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)
33. 33.
G. Psihoyios, T.E. Simos, Trigonometrically fitted predictor–corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003)
34. 34.
T.E. Simos, I.T. Famelis, Ch. Tsitouras, Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)
35. 35.
T.E. Simos, Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)
36. 36.
K. Tselios, T.E. Simos, Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math. 175(1), 173–181 (2005)
37. 37.
T.E. Simos, Ch. Tsitouras, Fitted modifications of classical Runge-Kutta pairs of orders 5(4). Math. Methods Appl. Sci. 41(12), 4549–4559 (2018)
38. 38.
Ch. Tsitouras, T.E. Simos, Trigonometric fitted explicit Numerov type method with vanishing phase-lag and its first and second derivatives. Mediterr. J. Math. 15(4), Article No. 168, (2018).
39. 39.
Z.A. Anastassi, T.E. Simos, An optimized Runge–Kutta method for the solution of orbital problems. J. Comput. Appl. Math. 175(1), 1–9 (2005)
40. 40.
T.E. Simos, Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems. Appl. Math. Lett. 22(10), 1616–1621 (2009)
41. 41.
S. Stavroyiannis, T.E. Simos, Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59(10), 2467–2474 (2009)
42. 42.
T.E. Simos, Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Applicandae Mathematicae 110(3), 1331–1352 (2010)
43. 43.
T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration, abstract and applied analysis, Vol. 2012, Article ID 182536, 15 p, 2012.
44. 44.
T.E. Simos, Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag, Journal of Applied Mathematics, Volume 2012, Article ID 420387, 17 pages, , (2012)
45. 45.
I. Alolyan, T.E. Simos, A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 53(8), 1915–1942 (2015)
46. 46.
I. Alolyan, T.E. Simos, Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(8), 1808–1834 (2015)
47. 47.
I. Alolyan, T.E. Simos, A high algebraic order predictor-corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 53(7), 1495–1522 (2015)
48. 48.
I. Alolyan, T.E. Simos, A family of explicit linear six-step methods with vanished phase-lag and its first derivative. J. Math. Chem. 52(8), 2087–2118 (2014)
49. 49.
T.E. Simos, An explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem. 52(3), 833–855 (2014)
50. 50.
Ibraheem Alolyan, T.E. Simos, A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52(3), 917–947 (2014)
51. 51.
I. Alolyan, T.E. Simos, A predictor-corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(2), 685–717 (2015)
52. 52.
I. Alolyan, T.E. Simos, A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52(9), 2334–2379 (2014)
53. 53.
G.A. Panopoulos, T.E. Simos, A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. J. Math. Chem. 51(7), 1914–1937 (2013)
54. 54.
T.E. Simos, New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: Construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)
55. 55.
T.E. Simos, High order closed Newton–Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation. J. Math. Chem. 50(5), 1224–1261 (2012)
56. 56.
D.F. Papadopoulos, A modified Runge–Kutta–Nyström method by using phase lag properties for the numerical solution of orbital problems. Appl. Math. Inf. Sci. 7(2), 433–437 (2013)
57. 57.
Th Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7(1), 81–85 (2013)
58. 58.
G.A. Panopoulos, T.E. Simos, An optimized symmetric 8-step semi-embedded predictor–corrector method for IVPs with oscillating solutions. Appl. Math. Inf. Sci. 7(1), 73–80 (2013)
59. 59.
D.F. Papadopoulos, T.E Simos, The use of phase lag and amplification error derivatives for the construction of a modified Runge–Kutta–Nyström method, abstract and applied analysis Article Number: 910624 Published: 2013Google Scholar
60. 60.
I. Alolyan, Z.A. Anastassi, T.E. Simos, A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218(9), 5370–5382 (2012)Google Scholar
61. 61.
I. Alolyan, T.E. Simos, A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 62(10), 3756–3774 (2011)
62. 62.
Ch. Tsitouras, ITh Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)
63. 63.
Ch. Tsitouras, ITh Famelis, T.E. Simos, Phase-fitted Runge–Kutta pairs of orders. J. Comput. Appl. Math. 321, 226–231 (2017)
64. 64.
T.E. Simos, Ch. Tsitouras, Evolutionary generation of high order, explicit two step methods for second order linear IVPs. Math. Methods Appl. Sci. 40, 6276–6284 (2017)
65. 65.
T.E. Simos, Ch. Tsitouras, A new family of 7 stages, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 40, 7867–7878 (2017)
66. 66.
D.B. Berg, T.E. Simos, Ch. Tsitouras, Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 41, 1845–1854 (2018)
67. 67.
T.E. Simos, Ch. Tsitouras, ITh Famelis, Explicit numerov type methods with constant coefficients: a review. Appl. Comput. Math. 16(2), 89–113 (2017)Google Scholar
68. 68.
A.A. Kosti, Z.A. Anastassi, T.E. Simos, Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems. Comput. Math. Appl. 61(11), 3381–3390 (2011)
69. 69.
Z. Kalogiratou, Th. Monovasilis, T.E. Simos, New modified Runge–Kutta–Nystrom methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60(6), 1639–1647 (2010)
70. 70.
Th. Monovasilis, Z. Kalogiratou, T.E. Simos, A family of trigonometrically fitted partitioned Runge-Kutta symplectic methods. Appl. Math. Comput. 209(1), 91–96 (2009)Google Scholar
71. 71.
Th. Monovasilis, Z. Kalogiratou, T.E. Simos, Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math. 13(4), 2271–2285 (2016)
72. 72.
Th. Monovasilis, Z. Kalogiratou, H. Ramos, T.E. Simos, Modified two-step hybrid methods for the numerical integration of oscillatory problems. Math. Methods Appl. Sci. 40(4), 5286–5294 (2017)
73. 73.
T.E. Simos, Multistage symmetric two-step P-stable method with vanished phase-lag and its first, second and third derivatives. Appl. Comput. Math. 14(3), 296–315 (2015)Google Scholar
74. 74.
Z. Kalogiratou, Th. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)
75. 75.
H. Ramos, Z. Kalogiratou, Th. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72, 1089–1102 (2016)Google Scholar
76. 76.
T.E. Simos, High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209(1), 137–151 (2009)Google Scholar
77. 77.
A. Konguetsof, T.E. Simos, An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Comput. Math. Appl. 45(1–3), 547–554 Article Number: PII S0898-1221(02)00354-1 (2003)
78. 78.
T.E. Simos, A new explicit hybrid four-step method with vanished phase-lag and its derivatives. J. Math. Chem. 52(7), 1690–1716 (2014)
79. 79.
T.E. Simos, On the explicit four-step methods with vanished phase-lag and its first derivative. Appl. Math. Inf. Sci. 8(2), 447–458 (2014)
80. 80.
G.A. Panopoulos, T.E. Simos, A new optimized symmetric embedded predictor–corrector method (EPCM) for initial-value problems with oscillatory solutions. Appl. Math. Inf. Sci. 8(2), 703–713 (2014)
81. 81.
G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 290, 1–15 (2015)
82. 82.
F. Hui, T.E. Simos, A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)
83. 83.
LGr Ixaru, M. Rizea, Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)
84. 84.
LGr Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
85. 85.
LGr Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)
86. 86.
J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge–Kutta–Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)
87. 87.
J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)
88. 88.
A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)
89. 89.
M.M. Chawla, P.S. Rao, An Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems II explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)
90. 90.
M.M. Chawla, P.S. Rao, An explicit sixth-order method with phase-lag of order eight for $$y^{\prime \prime }=f(t, y)$$. J. Comput. Appl. Math. 17, 363–368 (1987)Google Scholar
91. 91.
T.E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46, 981–1007 (2009)
92. 92.
A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)
93. 93.
A.D. Raptis, J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 36, 113–119 (1985)
94. 94.
R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. R. Soc. Ser. A 274, 427–442 (1963)
95. 95.
R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)
96. 96.
T.E. Simos, Exponentially fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation and related problems. Comput. Mater. Sci. 18, 315–332 (2000)
97. 97.
J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)
98. 98.
K. Mu, T.E. Simos, A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53, 1239–1256 (2015)
99. 99.
M. Liang, T.E. Simos, A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation. J. Math. Chem. 54(5), 1187–1211 (2016)
100. 100.
X. Xi, T.E. Simos, A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 54(7), 1417–1439 (2016)
101. 101.
F. Hui, T.E. Simos, Hybrid high algebraic order two-step method with vanished phase-lag and its first and second derivatives. MATCH Commun. Math. Comput. Chem. 73, 619–648 (2015)Google Scholar
102. 102.
Z. Zhou, T.E. Simos, A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54, 442–465 (2016)
103. 103.
F. Hui, T.E. Simos, Four stages symmetric two-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives. Appl. Comput. Math. 15(2), 220–238 (2016)Google Scholar
104. 104.
W. Zhang, T.E. Simos, A high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation. Mediterr. J. Math. 13(6), 5177–5194 (2016)
105. 105.
L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. 2016 Article ID 8181927, 20 p. Google Scholar
106. 106.
M. Dong, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat 31(15), 4999–5012 (2017)
107. 107.
R. Lin, T.E. Simos, A two-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Open Phys. 14, 628–642 (2016)
108. 108.
H. Ning, T.E. Simos, A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J. Math. Chem. 53(6), 1295–1312 (2015)
109. 109.
Z. Wang, T.E. Simos, An economical eighth-order method for the approximation of the solution of the Schrödinger equation. J. Math. Chem 55, 717–733 (2017)
110. 110.
J. Ma, An efficient and computational effective method for second order problems. J. Math. Chem. 55, 1649–1668 (2017)
111. 111.
V.N. Kovalnogov, R.V. Fedorov, V.M. Golovanov, B.M. Kostishko, T.E. Simos, A four stages numerical pair with optimal phase and stability properties. J. Math. Chem 56(1), 81–102 (2018)
112. 112.
K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. (2018).
113. 113.
J. Fang, C. Liu, T.E. Simos, A hybrid finite difference pair with maximum phase and stability properties. J. Math. Chem. 56(2), 423–448 (2018)
114. 114.
J. Yao, T.E. Simos, New finite difference pair with optimized phase and stability properties. J. Math. Chem. 56(2), 449–476 (2018)
115. 115.
J. Zheng, C. Liu, A new two-step finite difference pair with optimal properties. J. Math. Chem. 56(3), 770–798 (2018)
116. 116.
X. Shi, T.E. Simos, New five-stages finite difference pair with optimized phase properties. J. Math. Chem. 56(4), 982–1010 (2018)
117. 117.
L.I.U. Chenglian, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem. 55(5), 1213–1235 (2018)Google Scholar
118. 118.
J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem. 56(6), 1567–1594 (2018)
119. 119.
K. Yan, T.E. Simos, New Runge-Kutta type symmetric two-step method with optimized characteristics. J. Math. Chem. (2018).
120. 120.
V.N. Kovalnogov, R.V. Fedorov, A.A. Bondarenko, T.E. Simos, New hybrid two-step method with optimized phase and stability characteristics. J. Math. Chem. (2018).

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Ruru Hao
• 1
• T. E. Simos
• 2
• 3
• 4
• 5
• 6
1. 1.School of Information EngineeringChang’an UniversityXi’anChina
2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
3. 3.Group of Modern Computational MethodsUral Federal UniversityYekaterinburgRussian Federation
4. 4.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
5. 5.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
6. 6.AthensGreece