6.2 Methods for Initial Value Problems in ODES
139
the predictor-corrector method. In this method, the solution at the new time
step is predicted using the explicit Euler method:
where the * indicates that this is not the final value of the solution a t tn+'.
Rather, the solution is corrected by applying the trapezoid rule using bL+,
to compute the derivative:
This method can be shown t o be second order accurate (the accuracy of the
trapezoid rule) but has roughly the stability of the explicit Euler method.
One might think that by iterating the corrector, the stability might be improved but this turns out not t o be the case because this iteration procedure
converges to the trapezoid rule solution only if At is small enough.
This predictor-corrector method belongs to the two-level family, for which
the highest accuracy possible is second order. For higher-order approximations one must use information at more points. The additional points may
be ones at which data has already been computed or points between tn and
tn+1 which are used strictly for computational convenience; the former are
called multipoint methods, the latter, Runge-Kutta methods. Here, we shall
present multipoint methods; Runge-Kutta methods are presented in the next
section.
The best known multipoint methods, the Adams methods, are derived
by fitting a polynomial to the derivatives a t a number of points in time.
If a Lagrange polynomial is fit to f (tn-,, bnern), f (tn-rn+17 bn-rnfl 1, ...,
f ( t n , r n ) , and the result is used to compute the integral in Eq. (6.2), we
obtain an explicit method of order m + 1; methods of this type are called
Adams-Bashforth methods. For the solution of partial differential equations,
only the lower order methods are used. The first order method is explicit
Euler while the second and third order methods are:
and
If data a t tn+] is included in the interpolation polynomial, implicit methods,
known as Adams-Moulton methods, are obtained. The first order method is
implicit Euler, the second order one is trapezoid rule and the third order
method is:
139
the predictor-corrector method. In this method, the solution at the new time
step is predicted using the explicit Euler method:
where the * indicates that this is not the final value of the solution a t tn+'.
Rather, the solution is corrected by applying the trapezoid rule using bL+,
to compute the derivative:
This method can be shown t o be second order accurate (the accuracy of the
trapezoid rule) but has roughly the stability of the explicit Euler method.
One might think that by iterating the corrector, the stability might be improved but this turns out not t o be the case because this iteration procedure
converges to the trapezoid rule solution only if At is small enough.
This predictor-corrector method belongs to the two-level family, for which
the highest accuracy possible is second order. For higher-order approximations one must use information at more points. The additional points may
be ones at which data has already been computed or points between tn and
tn+1 which are used strictly for computational convenience; the former are
called multipoint methods, the latter, Runge-Kutta methods. Here, we shall
present multipoint methods; Runge-Kutta methods are presented in the next
section.
The best known multipoint methods, the Adams methods, are derived
by fitting a polynomial to the derivatives a t a number of points in time.
If a Lagrange polynomial is fit to f (tn-,, bnern), f (tn-rn+17 bn-rnfl 1, ...,
f ( t n , r n ) , and the result is used to compute the integral in Eq. (6.2), we
obtain an explicit method of order m + 1; methods of this type are called
Adams-Bashforth methods. For the solution of partial differential equations,
only the lower order methods are used. The first order method is explicit
Euler while the second and third order methods are:
and
If data a t tn+] is included in the interpolation polynomial, implicit methods,
known as Adams-Moulton methods, are obtained. The first order method is
implicit Euler, the second order one is trapezoid rule and the third order
method is: