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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.

  1. Any a priori estimate to $u$ and its derivatives;
  2. Any numerical methods which has estimate for absolute error (e.g. finite difference etc)

Many thanks !

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1 Answer 1

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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.

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  • $\begingroup$ Many thanks for your comment. I am looking for a second order accuracy using a finite difference method. However, I cannot find an appropriate reference for that. One related work I found was (jstor.org/stable/2004001?seq=1#metadata_info_tab_contents), which uses fully explicit scheme and gave a first order accuracy in time. I bet there should be improvement since then. $\endgroup$
    – Kenneth Ng
    Apr 2, 2020 at 15:12
  • $\begingroup$ I found this link of which page 85-87 looks very useful. Even the research paper link goes in a lot of depth. But only read this if you have good understanding of Sobolev spaces etc. I will first look at link 1. $\endgroup$
    – angrwl
    Apr 2, 2020 at 15:41

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