Unformatted text preview: Using the triangular inequality
jdp j jyn ; yn+1j + jyn+1 ; y(tn)j:
n The last term on the right is the local error of the order p + 1 method (3.5.4b) and is
jdp j jyn ; yn+1j + jdp+1j:
The higher-order error term on the right may be neglected to get an error estimate of
jdp j jyn ; yn+1j:
n (3.5.5) Embedding, like Richardson's extrapolation, is also an expensive way of estimating errors.
If the number of Runge-Kutta stages s p, then embedding requires approximately
m(p + 1) additional function evaluations per step for a system of m ODEs.
The number of function evaluations can be substantially reduced by embedding the
p th-order method within an (s +1)-stage method of order p +1. For explicit Runge-Kutta
methods, the tableau of the (s + 1)-stage method would have the form
... ... cs+1 as+1 1 as+1 2
(Zero's on an above the diagonal in A are not shown.) Assuming that the p th-order
Runge-Kutta method has s stages, it would be required to have the form
c3 a31 a32
... ... ... . . .
cs as1 as2
as s 1
With this form, only one additio...
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- Spring '14