# The Runge-Kutta Methods

It always is possible to reduce this integration error by making the DT of your models smaller. As DT approaches zero, Euler's approximation approaches the exact solution. But as you make DT smaller, you also increase the number of calculations (and the time) required to conduct your simulation - clearly not a desirable outcome. Another way to reduce the integration error is to use a more sophisticated procedure for estimating the change in stocks over a given DT. The two Runge-Kutta algorithms give you a higher degree of sophistication.

To aid in the exposition of Runge-Kutta, a bit of notation will prove helpful. Let "x" represent any stock and "f(t,x)" represent the functional notation for flows that depend on time "t" and stock "x".

## 2nd-Order Runge-Kutta

Of the two Runge-Kutta methods, 2nd-order is the simpler. Basically, this algorithm uses two flow calculations within a given DT to create an estimate for the change in a stock over the DT. As with Euler's method, the solution of equations involves both an initialization and an iteration phase.

###### Initialization Phase (Runge-Kutta 2):

Step 1. Create a list of all stocks, flows and converters in required order of evaluation.

Step 2. Calculate initial values of all stocks, flows and converters (in order of evaluation).

Time = From Time

Stockt=0 = f(initial values, stocks, converters, flows)

Converters = f(stocks, converters, flows)

Flows = f(stocks, converters, flows)

###### Iteration Phase (Runge-Kutta 2):

Step 1.

(a) Estimate change in stocks over the interval DT. Use two points to create this estimate:

(1) F1 = dt * f(t, x) [the same as Euler's flow estimate]

(2) F2 = dt * f(t+dt, x + F1)

(3)
stock = 1/2 * (F1 + F2)

(b) Calculate new value for stocks based on this estimate.

Stockt = Stockt-dt +
stock

Step 2. Calculate new values for flows and converters (in order of evaluation).

Converters = f(stocks, converters, flows)

Flows = f(stocks, converters, flows)

Step 3. Update simulation time. Stop iteration when Time is >= simulation To Time.

Time = Time + dt

The intermediate calculations in Step 1 of the iteration phase enable the 2nd-order Runge-Kutta method to create a more accurate estimate for the change in stocks during a given DT. This estimate is based on an average of flow values computed at the beginning, and at the end, of the DT interval.

## 4th-order Runge-Kutta

4th-order Runge-Kutta works just like its 2nd-order counterpart, except that it uses four rather than two flow calculations within a given DT to create an estimate for the change in a stock over the DT. A weighted average of these calculations is used as the estimate for the change in the stock:

###### Initialization Phase (Runge-Kutta 4):

Step 1. Create a list of all stocks, flows and converters in required order of evaluation.

Step 2. Calculate initial values of all stocks, flows and converters (in order of evaluation).

Time = From Time

Stockt=0 = f(initial values, stocks, converters, flows)

Converters = f(stocks, converters, flows)

Flows = f(stocks, converters, flows)

###### Iteration phase (Runge-Kutta 4):

Step 1.

(a) Estimate change in stocks over the interval DT. Use four points to create this estimate:

(1) F1 = dt * f(t,x)

(2) F2 = dt * f(t+dt/2, x + 1/2F1)

(2) F3 = dt * f(t+dt/2, x + 1/2F2)

(4) F4 = dt * f(t+dt, x + F3)

(5)
stock = 1/6 * (F1 + 2F2 + 2F3 + F4)

(b) Calculate new value for Stocks based on this estimate.

Stockt = Stockt-dt +
stock

Step 2. Calculate new values for flows and converters (in order of evaluation).

Converters = f(stocks, converters, flows)

Flows = f(stocks, converters, flows)

Step 3. Update simulation time. Stop iteration when Time is >= simulation To Time.

Time = Time + dt

The intermediate calculations in the first step of the iteration phase are the key to the 4th-order Runge-Kutta method. F1 represents the rate of change of the stock, i.e., the flow, at time "t". F2 is a new calculation of the flow at 1/2 an interval into the future, t+dt/2. In order to calculate F2, F1 is used to update the stocks. These then are used to re-calculate the flows at t+dt/2. One-half of F1 is used in estimating F2 since the calculation is made over only 1/2 the interval. F3 is a new calculation of the flow at t+dt/2. To calculate F3, F2 is used to update the stocks, which then are used to re-calculate the flows at t+dt/2. Again, only 1/2 of F2 is used to update the stocks since the calculation is made over 1/2 an interval. F4 is a calculation of the flow at one full interval in the future, t+dt. To calculate F4, F3 is used as the flow over the full interval. The stocks are updated using F3, then F4 is estimated.

Finally, the four intermediate flow calculations are averaged together, yielding an estimate of the flow. It is this average flow that is used to update the stock before the next iteration begins.