## Exponential Growth

Population growth can be described by dN/dt=gN. The analytic solution is N(t)=N0*exp(gt), where N0 is the initial population and g is the growth rate. We study the numeric solution to this problem using Euler's Method.
For the data shown here, N0=1000, g=0.2/year, and the maximum time is 10 years. We vary the timestep.

The Matlab code used is here

Typical output in the the file output.log can be found here

h=1.0

h=0.1

h=0.05

h=0.0.25

 h Numerical Result Exact Result Error 1.0 6.1917364e+03 7.3890561e+03 1.1973197e+03 0.1 7.2446461e+03 7.3890561e+03 1.4440998e+02 0.05 7.3160179e+03 7.3890561e+03 7.3038247e+01 0.025 7.3523251e+03 7.3890561e+03 3.6730991e+01 0.001 7.3889083e+03 7.3890561e+03 1.4777774e-01

For a really small timestep, the error actually increases due when the timestep is decreased (Theorem 5.10 in the textbook). For example, with the above code, and tmax=0.0000001, we get:
 h Numerical Result Exact Result Error 0.00000002 1.0000000e+03 1.0000000e+03 5.6843419e-13 0.00000001 1.0000000e+03 1.0000000e+03 1.1368684e-13 0.000000005 1.0000000e+03 1.0000000e+03 2.1600499e-12 0.000000001 1.0000000e+03 1.0000000e+03 4.4337867e-12