Numerical integrators for stiff and highly oscillatory differential equations

Author:
Simeon Ola Fatunla

Journal:
Math. Comp. **34** (1980), 373-390

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559191-X

MathSciNet review:
559191

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Abstract | References | Similar Articles | Additional Information

Abstract: Some *L*-stable fourth-order explicit one-step numerical integration formulas which require no matrix inversion are proposed to cope effectively with systems of ordinary differential equations with large Lipschitz constants (including those having highly oscillatory solutions). The implicit integration procedure proposed in Fatunla [11] is further developed to handle a larger class of stiff systems as well as those with highly oscillatory solutions. The same pair of nonlinear equations as in [11] is solved for the stiffness/oscillatory parameters. However, the nonlinear systems are transformed into linear forms and an efficient computational procedure is developed to obtain these parameters. The new schemes compare favorably with the backward differentiation formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].

**[1]**V. AMDURSKY & A. ZIV,*On Numerical Treatment of Stiff, Highly Oscillatory Systems*, IBM Technical Report 015, IBM Israel Scientific Center, 1974.**[2]**V. AMDURSKY & A. ZIV,*On the Numerical Solution of Stiff Linear Systems of the Oscillatory Type*, IBM Technical Report 032, IBM Israel Scientific Center, 1975.**[3]**V. AMDURSKY & A. ZIV,*The Numerical Treatment of Linear Highly Oscillatory ODE Systems by Reduction to Nonoscillatory Types*, IBM Technical Report 039, IBM Israel Scientific Center, 1976.**[4]**G. BJUREL, G. DAHLQUIST, B. LINDBERG, S. LINDEN & L. ODEN,*Survey of Stiff Ordinary Differential Equations*, Computer Science Report NA 70.11, Royal Institute of Technology, Stockholm, Sweden, 1970.**[5]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**0159424**, https://doi.org/10.1090/S0025-5718-1964-0159424-9**[6]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[7]**W. H. Enright,*Second derivative multistep methods for stiff ordinary differential equations*, SIAM J. Numer. Anal.**11**(1974), 321–331. MR**0351083**, https://doi.org/10.1137/0711029**[8]**W. H. ENRIGHT, T. E. HULL & B. LINDBERG, ``Comparing numerical methods for stiff systems of ODEs,''*BIT*, v. 15, 1975, pp. 10-48.**[9]**S. O. FATUNLA, ``A new algorithm for numerical solutions of ODEs,''*Comput. Math. Appl.*, v. 2, 1976, pp. 247-253.**[10]**Simeon O. Fatunla,*A variable order one-step scheme for numerical solution of ordinary differential equations*, Comput. Math. Appl.**4**(1978), no. 1, 33–41. MR**0501922**, https://doi.org/10.1016/0898-1221(78)90005-6**[11]**Simeon Ola. Fatunla,*An implicit two-point numerical integration formula for linear and nonlinear stiff systems of ordinary differential equations*, Math. Comp.**32**(1978), no. 141, 1–11. MR**0474830**, https://doi.org/10.1090/S0025-5718-1978-0474830-0**[12]**Walter Gautschi,*Numerical integration of ordinary differential equations based on trigonometric polynomials*, Numer. Math.**3**(1961), 381–397. MR**0138200**, https://doi.org/10.1007/BF01386037**[13]**C. W. Gear,*The automatic integration of stiff ordinary differential equations.*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 187–193. MR**0260180****[14]**C. W. GEAR, ``Algorithm 407: DIFSUB for solution of ordinary differential equations,''*Comm. ACM*, v. 14, 1971, pp. 185-190.**[15]**L. W. Jackson and Surender Kumar Kenue,*A fourth order exponentially fitted method*, SIAM J. Numer. Anal.**11**(1974), 965–978. MR**0362926**, https://doi.org/10.1137/0711075**[16]**J. D. Lambert,*Nonlinear methods for stiff systems of ordinary differential equations*, Conference on the Numerical Solution of Differential Equations (Univ. of Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 75–88. Lecture Notes in Math., Vol. 363. MR**0426436****[17]**J. D. Lambert and I. A. Watson,*Symmetric multistep methods for periodic initial value problems*, J. Inst. Math. Appl.**18**(1976), no. 2, 189–202. MR**0431691****[18]**J. Douglas Lawson,*Generalized Runge-Kutta processes for stable systems with large Lipschitz constants*, SIAM J. Numer. Anal.**4**(1967), 372–380. MR**0221759**, https://doi.org/10.1137/0704033**[19]**Bengt Lindberg,*On smoothing and extrapolation for the trapezoidal rule*, Nordisk Tidskr. Informationsbehandling (BIT)**11**(1971), 29–52. MR**0281356****[20]**W. LINIGER & R. A. WILLOUGHBY,*Efficient Numerical Integration of Stiff Systems of Ordinary Differential Equations*, IBM Research Report RC 1970, IBM, Yorktown Heights, New York, 1969.**[21]**W. L. Miranker and G. Wahba,*An averaging method for the stiff highly oscillatory problem*, Math. Comp.**30**(1976), no. 135, 383–399. MR**0423817**, https://doi.org/10.1090/S0025-5718-1976-0423817-0**[22]**W. L. Miranker, M. van Veldhuizen, and G. Wahba,*Two methods for the stiff highly oscillatory problem*, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 257–273. MR**0657229****[23]**W. L. Miranker and M. van Veldhuizen,*The method of envelopes*, Math. Comp.**32**(1978), no. 142, 453–496. MR**0494952**, https://doi.org/10.1090/S0025-5718-1978-0494952-8**[24]**Robert D. Skeel and Antony K. Kong,*Blended linear multistep methods*, ACM Trans. Math. Software**3**(1977), no. 4, 326–345. MR**0461922**, https://doi.org/10.1145/355759.355762**[25]**Arthur David Snider and Gary Charles Fleming,*Approximation by aliasing with application to “Certaine” stiff differential equations*, Math. Comp.**28**(1974), 465–473. MR**0343637**, https://doi.org/10.1090/S0025-5718-1974-0343637-3**[26]**E. Stiefel and D. G. Bettis,*Stabilization of Cowell’s method*, Numer. Math.**13**(1969), 154–175. MR**0263250**, https://doi.org/10.1007/BF02163234

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559191-X

Keywords:
Stiffness and oscillatory parameters,
ordinary differential equations,
*L*-stable,
eigenvalues,
meshsize,
explicit,
implicit,
*A*-stable,
exponential fitting

Article copyright:
© Copyright 1980
American Mathematical Society