Nonlinear stability of general linear methods for neutral delay differential equations. (English) Zbl 1167.65046

Wang, Wan-Sheng; Li, Shou-Fu; Su, Kai

J. Comput. Appl. Math. 224, No. 2, 592-601 (2009).

The authors consider the numerical solutions of neutral delay differential equations in the general nonlinear form,
\[ \begin{cases} y'(t) = f(t,y(t), y(t-\tau), y'(t-\tau)), & t\geq 0,\\ y(t) = \phi(t), & t\leq 0. \end{cases} \]
The main purpose is to study the stability of general linear methods, such as backward differentiation formulae (BDF’s), extended and modified extended BDF’s, parallel multi-value hybrid methods, Runge-Kutta, multi-step Runge-Kutta. A general class of methods to which the results can be applied is described in section 3 of the paper, while in section 2 the main theoretical results attending to some required properties of the components of the differential equation are given.
Section 3 also contains many definitions and comments related to stability that are enlarged with the introduction of the new concepts of GS(p), GAS(p), and weak GAS(p) stability of general linear methods with linear interpolation. Many implications between these and other concepts of stability, as well as the relation with \((k,p,0)\)-algebraic stability are investigated, specially in section 4, where the main results and proofs of the paper are given.
The paper ends with some examples of implementation of multi-step Runge-Kutta methods and some numerical experiments to support the results of the work.

Reviewer: Josep J. Masdemont (Barcelona)

MSC:

65L20	Stability and convergence of numerical methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
34K40	Neutral functional-differential equations
34K50	Stochastic functional-differential equations
34K28	Numerical approximation of solutions of functional-differential equations (MSC2010)
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

nonlinear neutral delay differential equations; general linear methods; numerical stability; algebraic stability; backward differentiation formulae; multi-step Runge-Kutta methods; numerical experiments

References:

[1]	Bellen, A.; Guglielmi, N.; Zennaro, M., On the contractivity and asymptotic stability of systems of delay differential equations of neutral type, BIT, 39, 1-24 (1999) · Zbl 0917.65071
[2]	Bellen, A.; Zennaro, M., Numerical Methods for Delay Differential Equations (2003), Oxford University Press: Oxford University Press Oxford · Zbl 0749.65042
[3]	Bellen, A.; Guglielmi, N.; Zennaro, M., Numerical stability of nonlinear delay differential equations of neutral type, J. Comput. Appl. Math., 125, 251-263 (2000) · Zbl 0980.65077
[4]	Hairer, E.; Zennaro, M., On error growth function of Runge-Kutta methods, Appl. Numer. Math., 22, 205-216 (1996) · Zbl 0871.65071
[5]	Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0425.34048
[6]	Da Hu, G.; Mitsui, T., Stability analysis of numerical methods for systems of neutral delay-differential equations, BIT, 35, 504-515 (1995) · Zbl 0841.65062
[7]	Huang, C. M.; Fu, H. Y.; Fu Li, S.; Chen, G. N., Stability analysis of Runge-Kutta methods for non-linear delay differential equations, BIT, 39, 270-280 (1999) · Zbl 0930.65090
[8]	Huang, C. M., Linear stability of general linear methods for systems of neutral delay differential equations, Appl. Math. Lett., 14, 1017-1021 (2001) · Zbl 0989.65078
[9]	Huang, C. M.; Li, S. F.; Fu, H. Y.; Chen, G. N., Nonlinear stability of general linear methods for delay differential equations, BIT, 42, 2, 380-392 (2002) · Zbl 1009.65054
[10]	Kolmanovskii, V.; Myshkis, A., Introduction to the Theory and Applications of Functional Differential Equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0917.34001
[11]	Kuang, J. X.; Xiang, J. X.; Tian, H. J., The asymptotic stability of one-parameter methods for neutral differential equations, BIT, 34, 400-408 (1994) · Zbl 0814.65078
[12]	Fu Li, S., Theory of Computational Methods for Stiff Differential Equations (1997), Hunan Science and Technology Publisher: Hunan Science and Technology Publisher Changsha
[13]	Fu Li, S., Stability and B-convergence properties of multistep Runge-Kutta method, Math. Comput., 69, 1481-1504 (1999) · Zbl 0954.65067
[14]	Qiu, L.; Yang, B.; Kuang, J. X., The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations, Numer. Math., 81, 3, 451-459 (1999) · Zbl 0918.65061
[15]	Tian, H. J.; Kuang, J. X.; Qiu, L., The stability of linear multistep methods for linear systems of neutral differential equation, J. Comput. Math., 19, 125-130 (2001) · Zbl 0985.65092
[16]	Vermiglio, R.; Torelli, L., A stable numerical approach for implicit non-linear neutral delay differential equations, BIT, 43, 195-215 (2003) · Zbl 1030.65078
[17]	Wang, W. S.; Li, S., Stability analysis of nonlinear delay differential equations of neutral type, Math. Num. Sinica, 26, 303-314 (2004) · Zbl 1495.34103
[18]	Wang, W. S.; Zhang, Y.; Li, S. F., Nonlinear stability of one-leg methods for delay differential equations of neutral type, Appl. Numer. Math., 58, 122-130 (2008) · Zbl 1137.65052
[19]	Wang, W. S.; Li, S. F.; Su, K., Nonlinear stability of Runge-Kutta methods for neutral delay differential equations, J. Comput. Appl. Math., 214, 175-185 (2008) · Zbl 1144.65054
[20]	Zhang, C. J.; Zhou, S. Z., The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay differential equations, Science in China, 41, 504-515 (1998)
[21]	Zhang, C. J., Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations, J. Comput. Math., 20, 583-590 (2002) · Zbl 1018.65101
[22]	Zhang, C. J., NGP \(( \alpha )\)-stability of general linear methods for NDDEs, Comput. Math. Appl., 47, 1105-1113 (2004) · Zbl 1073.65075