An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

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An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

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ACM Transactions on Mathematical Software (TOMS) archive
Volume 6 , Issue 3 (September 1980) table of contents

Pages: 295 - 318

Year of Publication: 1980

ISSN:0098-3500

Authors

K. R. Jackson	Department of Computer Science, Yale University, New Haven, CT
R. Sacks-Davis	Department of Computer Science, Monash University, Clayton, Victoria, Australia

Publisher

ACM New York, NY, USA

Bibliometrics

Downloads (6 Weeks): 14, Downloads (12 Months): 76, Citation Count: 11

Additional Information:

references cited by index terms collaborative colleagues peer to peer

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REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	BRAYTON, R K., AND CONLEY, C.C. Some results on the stability and instability of the backward differentiation methods with non-umform time steps. Proc. Int. Symp Numerical Analysas, Dubhn, 1972, pp. 13-33
	2	BRAYTON, R K., GUSTAVSON, F.G., AND HACHTEL, G.D. A new efficient algorithm for solving dffTerential-algebralc systems using imphcit backward differentiation formulas. Proc. IEEE 60 (1972), 98-108.
	3	G. D. Byrne , A. C. Hindmarsh, A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations, ACM Transactions on Mathematical Software (TOMS), v.1 n.1, p.71-96, March 1975 [doi>10.1145/355626.355636]
	4	BY~tNE, G.D., HINDMARSH, A.C., JACKSON, K.R, ASD BROWN, H.G. A comparison of two ODE codes' GEAR and EPISODE Comput. Chem. Eng. 1 (1977), 133-147.
	5	BYRNZ, G.D., HINDMARSH, A.C., JACKSON, K.R., AND BROWN, H.G. Comparative test results for two ODE solvers--EPISODE and GEAR, Tech. Rep. ANL-77-19, Argonne National Lab., Argonne, Ill., 1977.
	6	ENRIGHT, W.H. Second derivative multistep methods for stiff orchnary differential equatlons. SIAM J Numer. Anal. 11 (1974), 321-331.
	7	ENRIGHT, W.H, AND HULL, T.E. Comparing numerical methods for the solution of stiff systems of ODE's arising m chemmtry. In Numerical Methods for Differential Systems, L. Lapidus and W.E. Schmser, Eds Academic Press, New York, 1976, pp. 45-66.
	8	NRIGHT, W.H, AND HULL, T.E. Test results on initial value methods for non-stiff ordinary differential equations. SIAM J. Numer. Anal 13 (1976), 944-961
	9	ENRIGHT, W.I'I., HULL, T.E., AND LINDBERG, B. Comparing numerical methods for stiff systems of ODE's. BIT 15 (1975), 10-48
	10	C. William Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall PTR, Upper Saddle River, NJ, 1971
	11	GEAR, C.W Asymptotic esttmatlon of errors and derivatives for the numerical solution of ordinary differential equations. Information Processing 74, North-Holland, Amsterdam, 1974, pp. 447--451.
	12	GEAR, C.W., AND TU, K.W. The effect of variable mesh size on the stability of multistep methods. SIAM J Numer. Anal. 11 (1974), 1025-1043.
	13	GEAR, C.W., Tu, K.W, AND WATANABE, D.S. The stability of automatic programs for numerical problems. In St~ff Dtfferentml Systems, R.A. Willoughby, Ed. Plenum Press, New York, 1974, pp. 111-121.
	14	HINDMARSH, A.C. GEAR: Ordinary differentml equation system solver. Tech. Rep. UCID-30001, Rev 3, Lawrence Lavermore Lab., Univ. California, Llvermore, Calif., 1974.
	15	HINDMARSH, A.C., AND BYRNE, G.D. EPISODE: An effective package for the integration of systems of ordinary differential equations. Tech. Rep. UCID-30112, Rev. 1, Lawrence Livermore Lab., Umv. Calffornm, Llvermore, Calif., 1977
	16	Kenneth Ronald Jackson, Variable stepsize, variable order integrand approximation methods for the numerical solution of ordinary differential equations., 1978
	17	KROGH, F.T. VODQ/SVDQ/DVDQ--Vaxlable order integrators for the numerical solutmn of ordinary differentmi equatmns. TU Doc. CP-2308, NPO-11643, Jet Propulsmn Lab, California Inst. Technol., Pasadena, Calif., 1969.
	18	SACKS-DAWS, R. Software for the solutmn of stiff ODE's based upon second derivative formulas. Tech. Rep. 145, Dep Computer Science, Techmon, Haifa, Israel, 1979.
	19	Arthur Edward Sedgwick, An effective variable-order variable-step adams method., 1973
	20	SHAMPISZ, L.F., AND GORDON, M.K. Computer Solutmn of Ordinary D~fferentml Equatmns. W.H. Freeman, San Francmco, Calif., 1975
	21	SHAMPINE, L.F., AND GORDON, M.K. Local error and variable order Adams codes. Appl. Math. Cornput. 1 (1975), 47-66.
	22	SHAMPINE, L.F., WATTS, H.A, AND DAVENPORT, S.M Solving non-stiff ordinary differential equations--The state of the art. SIAM Rev. 18 (1975), 376-411.
	23	Tu, K.W. Stability and convergence of general multistep ~and multivalue methods with variable step size. Ph D. Dissertation, Tech. Rep UIUCDCS-R-72-526, Dep. Computer Scmnce, Univ. Illinois at Urbana-Champaign, Urbana, Ill., 1972.

CITED BY 11

	Fred T. Krogh , Kris Stewart, Asymptotic (h\rightarrow\infty) Absolute Stability for BDFs Applied to Stiff Differential Equations, ACM Transactions on Mathematical Software (TOMS), v.10 n.1, p.45-57, March 1984
	Zhao Li , C.-J. Richard Shi, A quasi-newton preconditioned Newton-Krylov method for robust and efficient time-domain simulation of integrated circuits with strong parasitic couplings, Proceedings of the 2006 conference on Asia South Pacific design automation, January 24-27, 2006, Yokohama, Japan
	R. Sacks-Davis, Fixed Leading Coefficient Implementation of SD-Formulas for Stiff ODEs, ACM Transactions on Mathematical Software (TOMS), v.6 n.4, p.540-562, Dec. 1980
	Xiaoji Ye , Wei Dong , Peng Li , Sani Nassif, MAPS: multi-algorithm parallel circuit simulation, Proceedings of the 2008 IEEE/ACM International Conference on Computer-Aided Design, November 10-13, 2008, San Jose, California
	Christopher E. Kees , Cass T. Miller, C++ implementations of numerical methods for solving differential-algebraic equations: design and optimization considerations, ACM Transactions on Mathematical Software (TOMS), v.25 n.4, p.377-403, Dec. 1999
	J. A. F. Hittinger , M. R. Dorr , R. L. Berger , E. A. Williams, Simulating time-dependent energy transfer between crossed laser beams in an expanding plasma, Journal of Computational Physics, v.209 n.2, p.695-729, 1 November 2005
	Steven L. Lee , Carol S. Woodward , Frank Graziani, Analyzing radiation diffusion using time-dependent sensitivity-based techniques, Journal of Computational Physics, v.192 n.1, p.211-230, 20 November 2003
	Daniel R. Reynolds , Ravi Samtaney , Carol S. Woodward, A fully implicit numerical method for single-fluid resistive magnetohydrodynamics, Journal of Computational Physics, v.219 n.1, p.144-162, 20 November 2006
	Peter N. Brown , Dana E. Shumaker , Carol S. Woodward, Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration, Journal of Computational Physics, v.204 n.2, p.760-783, 10 April 2005
	Alan C. Hindmarsh , Peter N. Brown , Keith E. Grant , Steven L. Lee , Radu Serban , Dan E. Shumaker , Carol S. Woodward, SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Transactions on Mathematical Software (TOMS), v.31 n.3, p.363-396, September 2005
	Gopal K. Gupta , Ron Sacks-Davis , Peter E. Tescher, A review of recent developments in solving ODEs, ACM Computing Surveys (CSUR), v.17 n.1, p.5-47, March 1985

INDEX TERMS

Primary Classification:
G. Mathematics of Computing
G.1 NUMERICAL ANALYSIS
G.1.7 Ordinary Differential Equations
Subjects: Stiff equations

General Terms:
Reliability, Theory

Collaborative Colleagues:

K. R. Jackson: colleagues

R. Sacks-Davis: colleagues

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