I am looking for some references of numerical methods to solve the heat equation of the form $$u_t = \alpha u_{xx}, \ t>0, x\in [0,l]$$ with mixed Dirichlet and Robin condition: $$u(0,x) = 0, \ u_x(t,l) + \beta u(t,l) = g(t), \ u(t,0)= f(t).$$
I am looking for reference for the followings.
- Any a priori estimate to $u$ and its derivatives;
- Any numerical methods which has estimate for absolute error (e.g. finite difference etc)
Many thanks !
There are three ways which I know of solving this equation:
two dimensional finite differences method when you discretize both the set $[0,l]$ and also time. Then you apply Backward Euler or Crank-Nicolson Method to compute this approximate function.
Second is using fourier series and Sturm-Louville Theorem (Look up the book introduction to Numerical Analysis by Endri Suli). It treats this problem as an eigenvalue problem for functions and then try to find that function (which is known as eigenfunction)
Finite Element Method using Galerkin Method and the notes can be found here: \link{https://courses.maths.ox.ac.uk/node/view_material/48430} which goes into functional analysis.
I would say the simplest one is Finite Differences and there are many truncation, stability (rounding errors) bounds already available. If your solution is differentiable in the domain, I would say finite differences is probably the best. Fourier Series gives you an exact solution as a infinite series. Third one is probably the most beautiful method although it has a lot of theory.
Hope this helps.