So I applied the Crank Nicolson discretisation first before multiplying by
test functions and integrating. But then I am however confused how to handle a
term like *(T_{n}^3 * T_{n+1} , w_i)_omega*
(where *(A,B)_omega* is the usual notation for the integral over the spatial
region omega of the product A*B and *w_i* is my test function).
If I write *T_{n}* and *T_{n+1}* as a finite linear combinaison of w_i such
that (*T_{n} = sum_i([T_{n}]_i * w_i)*) then *T_{n}^3* is monstruous unless I
You're thinking of T_n^3 as a polynomial of particular high degree, but we
don't actually care about that. We just need to evaluate it at quadrature
points, and to do that you evaluate
T_n(x_q) = \sum [T_n]_j \varphi_j(x_q)
and then take whatever value that gives you to the third power. You never need
T_n^3 as a *function*, you just need to be able to evaluate it at quadrature
points, and so questions such as whether functions are orthogonal don't
actually matter.
1- Is this even mathematically sound ? I should be able to choose my basis
orthogonal as far as I can think. Can I then do that with deal.II ?
Think about it this way: You want to compute an integral
\int c(x) w_i(x) w_j(x)
where c(x) is a coefficient that in your case involves T_n^3. You won't in
general be able to find orthogonal basis functions w_k for these kind of
weights c(x). You just have to approximate the integral by quadrature.
2 - Is this the approach I should be following for handling such term or
should I rethink the time discretisation scheme ?
No. Just think of the old solution as a coefficient. You probably want to take
a look at step-15, which shares many of the characteristics of what you want
to do.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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