12. Runge-Kutta (RK4) numerical solution for Differential Equations

In the last section, Euler's Method gave us one possible approach for solving differential equations numerically.

The problem with Euler's Method is that you have to use a small interval size to get a reasonably accurate result. That is, it's not very efficient.

The Runge-Kutta Method produces a better result in fewer steps.

Runge-Kutta Method Order 4 Formula

Applications of RK4

Mechanics

Biology

  • Predator-prey models
  • Fisheries collapses
  • Drug delivery
  • Epidemic prediction

Physics

  • Climate change models
  • Ozone protection

Aviation

  • On-board computers
  • Aerodynamics

`y(x+h)` `=y(x)+1/6(F_1+2F_2+2F_3+F_4)`

where

`F_1=hf(x,y)`

`F_2=hf(x+h/2,y+F_1/2)`

`F_3=hf(x+h/2,y+F_2/2)`

`F_4=hf(x+h,y+F_3)`

Where does this formula come from?

Here's a brief background to the formula.

Let's look at an example to see how it works.

Example

Use Runge-Kutta Method of Order 4 to solve the following, using a step size of `h=0.1` for `0<=x<=1`.

`dy/dx=(5x^2-y)/e^(x+y)`

`y(0) = 1`

Step 1

Note: The following looks tedious, and it is. We'll use a computer (not calculator) to do most of the work for us. The following is here so you can see how the formula is applied.

We start with `x=0` and `y=1`. We'll find the `F` values first:

`F_1=hf(x,y)` ` = 0.1(5(0)^2-1)/e^(0+1)` ` = -0.03678794411`

For `F_2`, we need to know:

`x+h/2 = 0+0.1/2 = 0.05`, and

`y+F_1/2` ` = 1 + (-0.03678794411)/2` ` = 0.98160602794`

We substitute these into the `F_2` expression:

`F_2=hf(x+h/2,y+F_1/2)` ` = 0.1((5(0.05)^2-0.98160602794)/e^(0.05+0.98160602794))` `=-0.03454223937`

For `F_3`, we need to know:

`y+F_2/2` ` = 1 + (-0.03454223937)/2` ` = 0.98272888031`

So

`F_3=hf(x+h/2,y+F_2/2)` ` = 0.1((5(0.05)^2-0.98272888031)/e^(0.05+0.98272888031))` `=-0.03454345267`

For `F_4`, we need to know:

`y+F_3` ` = 1 -0.03454345267` ` = 0.96545654732`

So

`F_4=hf(x+h,y+F_3)` ` = 0.1((5(0.1)^2-0.96545654732)/e^(0.1+0.96545654732))` `= -0.03154393258`

Step 2

Next, we take those 4 values and substitute them into the Runge-Kutta RK4 formula:

`y(x+h)` `=y(x)+1/6(F_1+2F_2+2F_3+F_4)`

`=1+1/6( -0.03678794411` ` -\ 2xx0.03454223937` ` -\ 2 xx0.03454345267` `{: -\ 0.03154393258)`

`=0.9655827899`

Using this new `y`-value, we would start again, finding the new `F_1`, `F_2`, `F_3` and `F_4`, and substitute into the Runge-Kutta formula.

We continue with this process, and construct the following table. (I used a spreadsheet to obtain the table. Using calculator is very tedious, and error-prone.)

The table is super wide, so I have put it in the following link:

Here is the graph of the solutions we found, from `x=0` to `x=1`.

Runge Kutta Order 4 solution of DE - graph solution

For interest, I extended the result up to `x=10`, and here is the resulting graph:

Runge Kutta Order 4 solution of DE - extended graph solution

Exercise

Solve the following using RK4 (Runge-Kutta Method of Order 4) for `0<=x<=2`. Use a step size of `h=0.2`:

`dy/dx=(x+y)sin xy`

`y(0) = 5`

Here's the table of values we get. Once again, it's a very wide table so I've put it separately here:

Here is the graph of the solutions we found, from `x=0` to `x=2`.

Runge Kutta Order 4 solution of DE - graph solution

For interest, I extended the result up to `x=6`, and here is the resulting graph:

Runge Kutta Order 4 solution of DE - extended graph solution

We observe the graph is not very smooth. If we use a smaller `h` value, the curve will generally be smoother.

In fact, this curve is asymptotic to the `x`-axis (it smoothly gets closer to the axis as `x` gets larger). The above graph shows what happens when our intervals are too coarse. (Those wiggles after around `x>4.5` should not be there.)

Here is the solution graph again, but this time I've used `h=0.1`. It's better, but still has a "wiggle" near `x=5` that should not be there.

Runge Kutta Order 4 solution of DE - smaller h solution

If I took even smaller values of `h`, I'd get a smoother curve. However, that's only up to a point, because rounding errors become significant eventually. Also, computing time goes up for little added benefit.

Improvements to Runge-Kutta

As you can see from the above example, there are points in the curve where the `y` values change relatively quickly (between `0` and `1`) and other places where the curve is more nearly linear (from `1` to `2`). Also, we have strange behavior near `x=5`.

In practice, when writing a computer program to perform Runge-Kutta, we allow for variable `h` values - quite small when the curve is changing quickly, and larger `h` values when the curve is relatively smooth.

Mathematics computer algebra systems (like Mathcad, Mathematica and Maple) include routines that calculate RK4 for us, so we just need to provide the function and the interval of interest.

Caution

Always be wary of your answers! Numerical solutions are only ever approximations, and under certain conditions, you can get chaotic results.

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