A modified Butcher formula for integration of stiff systems of ordinary differential equations

Author:
H. Nosrati

Journal:
Math. Comp. **27** (1973), 267-272

MSC:
Primary 65L05

MathSciNet review:
0329265

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Abstract | References | Similar Articles | Additional Information

Abstract: An *A*-stable one-step integration formula, called the modified Butcher (MB), is presented and is shown to have an order of accuracy , when the differential system is linear, and otherwise. A method for evaluating the local truncation error of the formula is also suggested. Finally, the main features of this formula, vis-a-vis the trapezoidal, are compared.

**[1]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[2]**C. W. Gear,*The Numerical Integration of Stiff Ordinary Differential Equations*, Report #221, Dept. Comp. Sci., University of Illinois, Urbana, Ill., January 1967.**[3]**Y. Genin,*A New Approach to the Synthesis of Stiffly Stable Linear Multistep Formulas*, Report #R188, MBLE Research Laboratory, Brussels, March 1972.**[4]**Werner Liniger and Ralph A. Willoughby,*Efficient integration methods for stiff systems of ordinary differential equations*, SIAM J. Numer. Anal.**7**(1970), 47–66. MR**0260181****[5]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**0159424**, 10.1090/S0025-5718-1964-0159424-9**[6]**Byron L. Ehle,*High order 𝐴-stable methods for the numerical solution of systems of D.E.’s*, Nordisk Tidskr. Informationsbehandling (BIT)**8**(1968), 276–278. MR**0239762****[7]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[8]**C. G. Broyden,*A class of methods for solving nonlinear simultaneous equations*, Math. Comp.**19**(1965), 577–593. MR**0198670**, 10.1090/S0025-5718-1965-0198670-6**[9]**C. G. Broyden,*A new method of solving nonlinear simultaneous equations*, Comput. J.**12**(1969/1970), 94–99. MR**0245197**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1973-0329265-3

Keywords:
Stiff system of ordinary differential equations,
*A*-stability,
Newton-Raphson iterations,
Butcher's implicit Runge-Kutta process

Article copyright:
© Copyright 1973
American Mathematical Society