David,

Sorry that I missed the attachment.

This whole issue is exactly why I tried to do this in Mathematica to see if there were differences and there are. I wanted the most accurate answer I could get and Mathematica gives me that.

That Euler notebook in Mathematica looks complicated. I have some studying to do there!

I am satisfied my original question has been answered thanks to both Bill and you!

Scott

David,

This is beautiful work and what I have been after for about 5 days now! To me it seems clear from your two simulations using Euler's method that one never knows when the time increment is small enough to ensure accuracy. One also runs into computational time problems in Excel with really, really small time increments. I guess I would have to go to 10 million rows in Excel to get the accuracy I need. Thank you again for all the time you put into solving this conundrum for me.

Do you mind sending me your notebook where you implemented Euler's method? I would like to know how to do that method in Mathematica for comparison purposes.

Scott

Hi Scott,

I implemented Euler's method in Mathematica using a function called NestList. It repeatedly applies a function to the last result, starting with some initial condition. I used your equations for the derivatives and your time step. The original result that both Bill and I have using NDSolve is this:

The result using Euler's method is this:

You can see that they are quite similar. And they converge on the same concentrations. I suspect the difference arises from the fixed time steps being to large in the early part of the simulation, resulting in under damping. Mathematica has adaptive time stepping to prevent that.

I don't know why the Excel Euler's method produces different results, but I would trust NDSolve over the perhaps-too-simple Euler's method.