Runge-Kutta-Fehlberg Method

Runge-Kutta-Fehlberg Method (RKF45)

GATEGATEGATEGATEOne way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice using step sizes h and
and compare answers at the mesh points corresponding to the larger step size.GATEGATEBut this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the agreement is not good enough. The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve this problem.GATEGATEIt has a procedure to determine if the proper step size h is being used.GATEGATEAt each step, two different approximations for the solution are made and compared.GATEGATEIf the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced.GATEGATEIf the answers agree to more significant digits than required, the step size is increased.

GATEGATEGATEGATEEach Runge-Kutta-Fehlberg step requires the use of the following six values:



GATEGATEGATE

Then an approximation to the solution of the I.V.P. is made using a Runge-Kutta method of order 4:



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And a better value for the solution is determined using a Runge-Kutta method of order 5:




The optimal step size sh can be determined by multiplying the scalar s times the current step size h. The scalar s is




where
is the specified error control tolerance.

Mathematica Subroutine (Runge-Kutta-Fehlberg Method RFK45)GATEGATETo compute a numerical approximation for the solution of the initial value problem
with
over
at a discrete set of points with an error controlGATEGATE
.GATEGATEStart with the initial pointGATEGATE
GATEGATEand generate the sequence of approximationsGATEGATE
.GATEGATEThe step size is automatically adjusted.GATEGATEThus the number
is determined by the subroutine.GATEGATEAt each step the final answer is computed using the formulaGATEGATEGATE
.GATEGATE

Remark.GATEGATEThe formula for adjusting the step size is a popular choice for numerical analysis textbooks.GATEGATE

Remark.GATEGATESome textbooks are now promoting the use of the RK order five formulaGATEGATE

GATEGATEas the accepted value instead ofGATEGATE
GATEGATEfrom the order four formula.GATEGATEThe following subroutine includes these adjustments.GATEGATE

GATE

Mathematica Subroutine (Runge-Kutta-Fehlberg Method RFK54)GATEGATETo compute a numerical approximation for the solution of the initial value problem
with
over
at a discrete set of points with an error controlGATEGATE
.GATEGATEStart with the initial pointGATEGATE
GATEGATEand generate the sequence of approximationsGATEGATE
.GATEGATEThe step size is automatically adjusted.GATEGATEGATEThus the number
is determined by the subroutine.GATEGATEAt each step the final answer is computed using the formula

.GATEGATE

Remark.GATEGATEOne might question the wisdom of using the RKF54 instead of the RKF45, it is still being debated.GATEGATEIn some sense the "guarantee" of accuracy is diminished.GATEGATEThe RKF54 is given for you to explore.