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Numerical Methods To Solve Initial
Value Problems
An Over View of Runge-Kutta Fehlberg and
Dormand and Prince Methods.
William Mize
Quick Refresher
ο We are looking at Ordinary Differential Equations
ο More specifically Initial Value Problems
ο Simple Examples:
xβ² = π₯ + 1
ο
Solution of: π₯ = π π‘ β 1
π₯ 0 =0
ο
x β² = 6π‘ β 1
Solution of: π₯ = 3π‘ 2 β t + 4
π₯ 1 =6
A Problem
ο How practical are analytical methods?
β²
ο Equation: x = π
β π‘ 2 βπ πππ‘
+ ln |π πππ‘ + π‘ππβπ‘ 3 |
ο We chose to find a Numerical solution because
ο Closed-form is to difficult to evaluate
ο No close-form solution
Some Quick Ground work
ο First Start with Taylor Series
Approximations
ο Then Move onto Runge-Kutta Methods for
Approximations
ο Lastly onto Runge-Kutta Fehlberg and
Dormand and Prince Methods for
Approximation and keeping control of error
How these Methods Work
ο All of the Methods will be using a step size
method.
ο Error is determined by the size of step,
order, and method used.
ο When actually calculating these, almost
always done via computer.
Taylor Series Methods(Brief)
ο Taylor Series As Follows
ο x π‘ + β = π₯ π‘ + βπ₯ β² π‘ +
1 2 β²β²
β π₯
2!
π‘ +
1 3 β²β²β² π‘
β π₯
3!
+
ο Most Basic is Eulerβs Method
ο x π‘ + β β π₯ π‘ + βπ₯ β² π‘
ο Higher Order Approximations better Accuracy
ο But at a cost
ο What can we do?
1 4 β²β²β²β²
β π₯
4!
π‘ +. .
Runge-Kutta Methods
ο Named After Carl Runge and Wilhelm Kutta
ο What they do?
ο Do the same Job as Taylor Series Method, but
without the analytic differentiation.
ο Just like Taylor Series with higher and higher
order methods.
ο Runge-Kutta Method of Order 4 Well accepted
classically used algorithm.
Runge-Kutta of Order 2
ο We donβt want to take derivatives for approximations
ο Instead use Taylor series to create Runge-Kutta methods to
approximate solution with just function evaluations.
β
ο π 2 π‘, π¦ = π π‘, π¦ + π β² π‘, π¦
2
ο We Want to Approximate this with
ο π΄π π‘ + π, π₯ + π
ο Find A, B, C
ο We get:
πΎ1 = βπ(π‘, π₯)
ο
πΎ2 = βπ(π‘ + β, π₯ + πΎ1 )
ο
1
2
x π‘ + β = π₯ π‘ + (πΎ1 + πΎ2 )
Error π(β2 )
Runge-Kutta of Order 4
1
x π‘ + β = π₯ π‘ + (πΎ1 + 2πΎ2 + 2πΎ3 + πΎ4 )
6
πΎ1 = βπ(π‘, π₯)
1
1
πΎ2 = βπ(π‘ + β, π₯ + πΎ1 )
2
2
1
1
πΎ3 = βπ(π‘ + β, π₯ + πΎ2 )
2
2
πΎ4 = βπ(π‘ + β, π₯ + πΎ3 )
Error of Order π(β5 )
So What's next?
ο Already Viable Numerical Solution established what's the
next step?
ο We want to control our Error and Step size at each step.
ο These methods are called adaptive.
ο Why?
ο Cost Less
ο Keep within Tolerance
ο Also look for More efficient ways of doing these things.
ο 10 Function Evaluation for RK4 and RK5
ο Just 6 for RKF4(5)
Runge-Kutta Fehlberg
ο Coefficients β πΎ , πΆπΎ , Ξ²πΎΞ» , πΆπΎ are found via Taylor
expansions
Next Step to find These Coefficients
Further Deriving
ο We assume πΆ1 = 0, πΆ1 = 0, β 4 =1
More and moreβ¦
ο So this was way more complicated than I actually
thought it would be.
ο But itβs all leading some where!
ο Eventually we want to have all the πΆπΎ , Ξ²πΎΞ» , πΆπΎ
in terms of β 2 and β 5 .
ο From there was must figure out our β
β 5.
ο β 5 ends up being arbitary
2 and
How to find β
2
ο
First Take coefficients from the 5th order equation.
ο
Which ultimately leads to
ο
Where we chose β
2=
1/3 and β
2=
3/8
β
2=
1/3
β
2=
3/8
Comparison(Problem)
Comparisons of Methods
Dormand and Prince Methods
Visual Comparison of Methods
Conclusion
ο Taylorβs method uses derivatives to solve ODE
ο RK uses only a combination of specific function evaluations
instead of derivatives to approximate solution of the ODE
ο RKF is beneficial because you can control your step size so
you have your global error within a predetermined tolerance
ο RK4 and RK5 uses 10 function evaluations vs RKF just 6
ο Runge-Kutta Fehlberg is widely accepted and used
commercially(Matlab, Mathematica, maple, etc)
Sources
ο Numerical Mathematics and Computing. Sixth Edition; Ward
Cheny, David Kincaid
ο Low-Order classical Runge-Kutta Formulas with StepSize
Control and their Application to some heat transfer
problems. By Erwin Fehlberg(1969)
ο A family of embedded Runge-Kutta Formulae. By Dormand
and Prince(1980)