Dormand and Prince Methods Numerical Methods To Solve Initial Value Problems William Mize Quick Refresher We are looking at Ordinary Differential Equations More specifically Initial Value Problems ID: 673474
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An Over View of Runge-Kutta Fehlberg and Dormand and Prince Methods.
Numerical Methods To Solve Initial Value Problems
William MizeSlide2
Quick Refresher
We are looking at Ordinary Differential EquationsMore specifically Initial Value Problems
Simple Examples:
Solution of: Solution of:
Slide3
A Problem
How practical are analytical methods?Equation:
We chose to find a Numerical solution because
Closed-form is to difficult to evaluate
No close-form solution Slide4
Some Quick Ground work
First Start with Taylor Series ApproximationsThen Move onto Runge-Kutta Methods for ApproximationsLastly onto Runge-Kutta Fehlberg and
Dormand
and Prince
Methods for Approximation and keeping control of errorSlide5
How these Methods Work
All of the Methods will be using a step size method.Error is determined by the size of step, order, and method used.When actually calculating these, almost always done via computer.Slide6
Taylor Series Methods(Brief)
Taylor Series As Follows
Most Basic is Euler’s Method
Higher Order Approximations better Accuracy
But at a cost
What can we do?
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Runge-Kutta Methods
Named After Carl Runge and Wilhelm Kutta
What they do?
Do the same Job as Taylor Series Method, but without the analytic differentiation.
Just like Taylor Series with higher and higher order methods.Runge-Kutta Method of Order 4 Well accepted classically used algorithm.Slide8
Runge-Kutta of Order 2
We don’t want to take derivatives for approximations
Instead use Taylor series to create Runge-Kutta methods to approximate solution with just function evaluations.
We Want to Approximate this with
Find A, B, CWe get:
Error
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Runge-Kutta of Order 4
Error of Order
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So What's next?
Already Viable Numerical Solution established what's the next step?We want to control our Error and Step size at each step.These methods are called adaptive.
Why?
Cost Less
Keep within ToleranceAlso look for More efficient ways of doing these things.10 Function Evaluation for RK4 and RK5Just 6 for RKF4(5)Slide11
Runge-Kutta Fehlberg
Coefficients
are found via Taylor expansions
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Next Step to find These CoefficientsSlide13
Further Deriving
We assume
=1
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More and more…
So this was way more complicated than I actually thought it would be.
But it’s all leading some where!
Eventually we want to have all the
in terms of From there was must figure out our and
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How to find
First Take coefficients from the 5
th
order equation.Which ultimately leads to Where we chose = 1/3 and = 3/8 Slide16
= 1/3
Slide17
=
3/8
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Comparison(Problem)Slide19
Comparisons of MethodsSlide20
Dormand and Prince MethodsSlide21
Visual Comparison of MethodsSlide22
Conclusion
Taylor’s method uses derivatives to solve ODERK uses only a combination of specific function evaluations instead of derivatives to approximate solution of the ODERKF is beneficial because you can control your step size so you have your global error within a predetermined tolerance
RK4 and RK5 uses 10 function evaluations
vs
RKF just 6Runge-Kutta Fehlberg is widely accepted and used commercially(Matlab, Mathematica, maple, etc)Slide23
Sources
Numerical Mathematics and Computing. Sixth Edition; Ward Cheny, David KincaidLow-Order classical Runge-Kutta Formulas with
StepSize
Control and their Application to some heat transfer problems. By Erwin
Fehlberg(1969)A family of embedded Runge-Kutta Formulae. By Dormand and Prince(1980)