Numerical Methods To Solve Initial
Value Problems
An Over View of Runge-Kutta Fehlberg and
Dormand and Prince Methods.
William Mize
Quick Refresher
 We are looking at Ordinary Differential Equations
 More specifically Initial Value Problems
 Simple Examples:
x′ = 𝑥 + 1

Solution of: 𝑥 = 𝑒 𝑡 − 1
𝑥 0 =0

x ′ = 6𝑡 − 1
Solution of: 𝑥 = 3𝑡 2 − t + 4
𝑥 1 =6
A Problem
 How practical are analytical methods?
′
 Equation: x = 𝑒
− 𝑡 2 −𝑠𝑖𝑛𝑡
+ ln |𝑠𝑖𝑛𝑡 + 𝑡𝑎𝑛ℎ𝑡 3 |
 We chose to find a Numerical solution because
 Closed-form is to difficult to evaluate
 No close-form solution
Some Quick Ground work
 First Start with Taylor Series
Approximations
 Then Move onto Runge-Kutta Methods for
Approximations
 Lastly onto Runge-Kutta Fehlberg and
Dormand and Prince Methods for
Approximation and keeping control of error
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.
Taylor Series Methods(Brief)
 Taylor Series As Follows
 x 𝑡 + ℎ = 𝑥 𝑡 + ℎ𝑥 ′ 𝑡 +
1 2 ′′
ℎ 𝑥
2!
𝑡 +
1 3 ′′′ 𝑡
ℎ 𝑥
3!
+
 Most Basic is Euler’s Method
 x 𝑡 + ℎ ≈ 𝑥 𝑡 + ℎ𝑥 ′ 𝑡
 Higher Order Approximations better Accuracy
 But at a cost
 What can we do?
1 4 ′′′′
ℎ 𝑥
4!
𝑡 +. .
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.
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.
ℎ
 𝑇 2 𝑡, 𝑦 = 𝑓 𝑡, 𝑦 + 𝑓 ′ 𝑡, 𝑦
2
 We Want to Approximate this with
 𝐴𝑓 𝑡 + 𝑏, 𝑥 + 𝑐
 Find A, B, C
 We get:
𝐾1 = ℎ𝑓(𝑡, 𝑥)

𝐾2 = ℎ𝑓(𝑡 + ℎ, 𝑥 + 𝐾1 )

1
2
x 𝑡 + ℎ = 𝑥 𝑡 + (𝐾1 + 𝐾2 )
Error 𝑂(ℎ2 )
Runge-Kutta of Order 4
1
x 𝑡 + ℎ = 𝑥 𝑡 + (𝐾1 + 2𝐾2 + 2𝐾3 + 𝐾4 )
6
𝐾1 = ℎ𝑓(𝑡, 𝑥)
1
1
𝐾2 = ℎ𝑓(𝑡 + ℎ, 𝑥 + 𝐾1 )
2
2
1
1
𝐾3 = ℎ𝑓(𝑡 + ℎ, 𝑥 + 𝐾2 )
2
2
𝐾4 = ℎ𝑓(𝑡 + ℎ, 𝑥 + 𝐾3 )
Error of Order 𝑂(ℎ5 )
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 Tolerance
 Also look for More efficient ways of doing these things.
 10 Function Evaluation for RK4 and RK5
 Just 6 for RKF4(5)
Runge-Kutta Fehlberg
 Coefficients ∝ 𝐾 , 𝐶𝐾 , β𝐾λ , 𝐶𝐾 are found via Taylor
expansions
Next Step to find These Coefficients
Further Deriving
 We assume 𝐶1 = 0, 𝐶1 = 0, ∝ 4 =1
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 𝐶𝐾 , β𝐾λ , 𝐶𝐾
in terms of ∝ 2 and ∝ 5 .
 From there was must figure out our ∝
∝ 5.
 ∝ 5 ends up being arbitary
2 and
How to find ∝
2

First Take coefficients from the 5th order equation.

Which ultimately leads to

Where we chose ∝
2=
1/3 and ∝
2=
3/8
∝
2=
1/3
∝
2=
3/8
Comparison(Problem)
Comparisons of Methods
Dormand and Prince Methods
Visual Comparison of Methods
Conclusion
 Taylor’s method uses derivatives to solve ODE
 RK uses only a combination of specific function evaluations
instead of derivatives to approximate solution of the ODE
 RKF 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 6
 Runge-Kutta Fehlberg is widely accepted and used
commercially(Matlab, Mathematica, maple, etc)
Sources
 Numerical Mathematics and Computing. Sixth Edition; Ward
Cheny, David Kincaid
 Low-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)