Stability of sequences generated by nonlinear differential systems

Author:
R. Leonard Brown

Journal:
Math. Comp. **33** (1979), 637-645

MSC:
Primary 65L99; Secondary 34D20

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521279-9

MathSciNet review:
521279

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Abstract: A local stability analysis is given for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. The standard linear stability analysis is reviewed, then it is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and of the one-leg *k*-step numerical solution can be approximated computationally. Correspondence between the one-leg *k*-step solution and its associated linear *k*-step solution is shown, and two examples are given.

**[1]**R. L. BROWN, "Graphical stability comparison of analytic and numerical solutions of nonlinear systems,"*Numerical Methods for Differential Equations and Simultation*, A. W. Bennett and R. Vichnevetsky (eds.), North-Holland, Amsterdam, 1978.**[2]**J. C. BUTCHER, "On the attainable order of Runge-Kutta methods,"*Math. Comp.*, v. 19, 1965, pp. 408-417. MR**0179943 (31:4180)****[3]**G. DAHLQUIST, "Numerical integration of ordinary differential equations,"*Math. Scand.*, v. 4, 1956, pp. 33-50. MR**0080998 (18:338d)****[4]**G. DAHLQUIST,*On Stability and Error Analysis for Stiff Non-Linear Problems*, Report NA-7508, Dept. of Information Processing, Royal Institute of Technology, Stockholm, 1975.**[5]**C. W. GEAR, "Algorithm 407: DIFSUB for solution of ordinary differential equations,"*Comm. ACM*, v. 14, 1971, pp. 185-190.**[6]**C. W. GEAR,*Numerical Initial Value Problems in Ordinary Differential Equations*, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR**0315898 (47:4447)****[7]**A. C. HINDMARSH,*GEAR*:*Ordinary Differential Equation System Solver*, UCID-300001, Rev. 2, Lawrence Livermore Lab., Livermore, California, 1972.**[8]**F. T. KROGH, "A variable step variable order multistep method for the numerical solution of ordinary differential equations,"*Information Processing*68, Vol. I, A. J. H. Morrell (ed.), North-Holland, Amsterdam, 1969, pp. 194-199. MR**0261790 (41:6402)****[9]**W. LINIGER & F. ODEH,*On Liapunov Stability of Stiff Non-Linear Multistep Difference Equations*, AFOSR-TR-76-1023, IBM Thomas J. Watson Research Center, 1976.**[10]**L. F. SHAMPINE & M. K. GORDON,*Computer Solution of Ordinary Differential Equations*:*Initial Value Problems*, Freeman, San Francisco, Calif., 1976. MR**0478627 (57:18104)****[11]**G. G. STEINMETZ, R. V. PARRISH & R. L. BOWLES,*Longitudinal Stability and Control Derivatives of a Jet Fighter Airplane Extracted from Flight Test Data by Utilizing Maximum Likelihood Estimation*, NASA-TV D-6532, NASA Langley Research Center, Hampton, Va., 1972.**[12]**K. W. TU,*Stability and Convergence of General Multistep and Multivalue Methods with Variable Stepsize*, UIUCDCS-R-72-526, Univ. of Illinois, Urbana, Ill., 1972.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521279-9

Keywords:
Initial value problem,
numerical integration of ordinary differential equations,
discrete Liapunov function,
stability of nonlinear sequences

Article copyright:
© Copyright 1979
American Mathematical Society