![[Maple Math]](images/lnotes1926.gif){kind=small}
at different (calculated) points rather then evaluations of derivatives of
![[Maple Math]](images/lnotes1927.gif){kind=small}
at the given point. To derive them we must first revise the following theorem:
Numerical Solutions for Ordinary Differential Equations
Preliminary Theory
Taylor Method(s)
Runge-Kutta Methods
The philosophy behind the Runge-Kutta methods is to emulate the local truncation errors of Taylor's method, using a combination of evaluations of
at different (calculated) points rather then evaluations of derivatives of
at the given point. To derive them we must first revise the following theorem:
Theorem 23 (Taylor's Theorem in two variables)
Midpoint method
Modified Euler's method
Heun's method
Fourth order Runge-Kutta
The next worksheet will look at the application of these methods and their accuracy.
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Now that we have an efficient method, can we possibly estimate the truncation error at each step and make sure it remains below a given threshold?
Error control
Runge-Kutta-Fehlberg Method
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Multistep Methods
Systems of ODE's
Boundary Value Problems