Numerical Analysis/Order of RK methods/Implicit RK2 on an Autonomous ODE

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We consider an autonomous initial value ODE

 

 

 

 

(ODE)

Applying the Tradezoidal rule gives the implicit Runge-Kutta method

 

 

 

 

(method)

We will show that (method ) is second order.

Expanding the true solution about using Taylor series, we have

Since satisfies (ODE ), we can substitute and obtain

 

 

 

 

(true)

In (method ) we can assume since that is the previous data. Subtracting (method ) from (true ) gives us the local truncation error

 

 

 

 

(error1)

In order to cancel more terms we need to expand . However, so we cannot do a regular Taylor expansion. Instead we can plug (method ) back into and then do a Taylor expansion to obtain

 

 

 

 

(implicit)

Substituting (implicit ) into (error1 ) yields

 

 

 

 

(error2)

This substitution was productive since the terms canceled. We can do this trick again, but this time only need (implicit ) up to since everything will be multiplied by at least and this can go into the . Substituting (implicit ) in for the first occurance of in (error2 ) yields

 

 

 

 

(error3)

This substitution was productive since the terms canceled. We can do this again, now truncating (implicit ) at . Substituting (implicit ) into (error3 ) yields

Since the term does not cancel, we have shown that the local truncation error is and thus the method is order 2.