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

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We consider an autonomous initial value ODEString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(ODE)

String Module Error: function rep expects a number as second parameter, received "

"

Applying the Tradezoidal rule

gives the implicit Runge-Kutta methodString Module Error: function rep expects a number as second parameter, received "

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(method)

String Module Error: function rep expects a number as second parameter, received "

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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 obtainString Module Error: function rep expects a number as second parameter, received "

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(true)

String Module Error: function rep expects a number as second parameter, received "

"

In (method ) we can assume since that is the previous data. Subtracting (method ) from (true

) gives us the local truncation errorString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(error1)

String Module Error: function rep expects a number as second parameter, received "

"

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 obtainString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(implicit)

String Module Error: function rep expects a number as second parameter, received "

"

Substituting (implicit ) into (error1

) yieldsString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(error2)

String Module Error: function rep expects a number as second parameter, received "

"

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

) yieldsString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(error3)

String Module Error: function rep expects a number as second parameter, received "

"

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.