Here we plot the behavior of the solution for 5 different choices of mu ... When the real part of mu is positive the solution grows exponentially ...
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Numerical Methods for Partial Differential Equations
Instructor: Tim Warburton
Here we plot the dependence of the solution to the top left ODE
on mu’s position in the complex plane
Here we plot the behavior of the solution for 5 different choices of mu
The error is quite small.
The error is visible but the trend is not wildly wrong.
The solution is not remotely correct – but is at least bounded.
Boom – the approximate solution grows exponentially fast, while the true solution decays!
two kinds of error can accumulate!.
Again, we have recovered Euler’s method.
Let’s choose f(u) = mu*u
We can solve this immediately:
Next suppose Re(mu)<=0 then we expect the actual
solution to be bounded in time by u(0).
For this to be true of the approximate solution werequire:
i) d small, c large
ii) d large, c small
of the f values
Last step: homework exercise