Unformatted text preview: ) ; (ftfy + ffy2)](t ;1 y(t ;1 ) + O(h4):
n n Clearly this is too complicated to be used as a practical error estimation scheme.
Two practical approaches to estimating the local and local discretization errors of
Runge-Kutta methods are (i) Richardson's extrapolation (or step doubling) and (ii) embedding. We'll study Richardson's extrapolation rst.
For simplicity, consider a scalar one-step method of order p having the following form
and local error yn = yn 1 + h (tn 1 yn 1 h) (3.5.2a) dn = Cnhp+1 + O(hp+2): (3.5.2b) ; ; ; The coe cient Cn may depend on tn 1 and y(tn 1) but is independent of h. Typically,
Cn is proportional to y(p+1)(tn 1). Of course, the ODE solution must have derivatives of
order p + 2 for this formula to exist.
Let yn be the solution obtained from (3.5.2a) using a step size h. Calculate a second
solution yn 2 at t = tn using two steps with a step size h=2 and an \initial condition"
of yn 1 at tn 1. (We'll refer to the solution computed at tn 1=2 = tn 1 + h=2 as yn 21=2.
Assuming that the error after two steps of size h=2 is twice that after one step (i.e.,...
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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations