A note on trapezoidal methods for the solution of initial value problems

Author:
A. R. Gourlay

Journal:
Math. Comp. **24** (1970), 629-633

MSC:
Primary 65.61

MathSciNet review:
0275680

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included.

**[1]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[2]**J. T. Day,*A Runge-Kutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations*, Comput. J.**9**(1966), 81–83. MR**0192665****[3]**M. K. Jain and K. D. Sharma,*Cubature method for the numerical solution of the characteristic initial value problem 𝑢_{𝑥𝑦}=𝑓(𝑥,𝑦,𝑢,𝑢ₓ,𝑢_{𝑦})*, J. Austral. Math. Soc.**8**(1968), 355–368. MR**0228202****[4]**Hans J. Stetter and W. Törnig,*General multistep finite difference methods for the solution of 𝑢_{𝑥𝑦}=𝑓(𝑥,𝑦,𝑢,𝑢ₓ,𝑢_{𝑦})*, Rend. Circ. Mat. Palermo (2)**12**(1963), 281–298. MR**0169394**

Retrieve articles in *Mathematics of Computation*
with MSC:
65.61

Retrieve articles in all journals with MSC: 65.61

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1970-0275680-3

Keywords:
Initial value problems,
stiff systems,
trapezoidal rule,
Goursat problem,
hyperbolic differential equations

Article copyright:
© Copyright 1970
American Mathematical Society