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A Error for perturbed solution vs Numeric

  1. Apr 8, 2017 #1

    joshmccraney

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    Gold Member

    Hi PF!

    I'm solving an PDE where the analytic solution is called ##F(x)## (unknown). To approximate the analytic solution I made a naive expansion in some small parameter ##\epsilon## such that ##F(x) = f_0(x)+\epsilon f_1(x)+O(\epsilon^2)##, where I know ##f_0(x)## and ##f_1(x)##. I then solved the PDE numerically, let's call that solution ##F_n##. Then the error ##(F_n - (f_0(x)+\epsilon f_1(x)))/F_n## should be ##O(\epsilon^2)##. However, when I let ##\epsilon=.9## and then ##\epsilon=.8## my error is still about ##0.15##. How can this be?

    I should say I know the numeric and asymptotic solutions are correct.
     
  2. jcsd
  3. Apr 11, 2017 #2

    Mark44

    Staff: Mentor

    ##\epsilon=.9## isn't really all that small. Do you get better results for, say, ##\epsilon=.1##?
     
  4. Apr 12, 2017 #3

    joshmccraney

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    Gold Member

    So for ##\epsilon=0.01## I'm getting an error of 0.165 but for ##\epsilon=0.2## the error is 0.14. How could this ever be possible?
     
  5. Apr 12, 2017 #4

    Mark44

    Staff: Mentor

    Maybe your approximate solution isn't that close to the actual solution...
     
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