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I'm interested in testing some algorithms on the heat equation, and I'd like to assess their accuracy. When evolving a Hamiltonian system, one has the energy to check the validity/correctness of the algorithm, which turns out to be (arguably) the most important quantity to check. Is there a similar quantity for the heat equation - i.e. a quantity which would very well characterize the error of the algorithm?

I could compute the time-evolution of the "energy" of the heat equation easily by the following

$$u_t(x,t)=\hat{H}u(x,t),~~~\hat{H}=\frac{\partial^2}{\partial x^2}$$

The "energy" would then be the expectation value of the $\hat{H}$ operator, which is simple to evaluate in Fourier space. However, I don't know if this "quantity" would even be useful.

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marked as duplicate by Kirill, Christian Clason, Paul Sep 16 '16 at 16:31

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  • $\begingroup$ The standard approach is the method of manufactured solutions; in fact, it's best to use that method even when there is a conserved quantity. $\endgroup$ – Kirill Sep 15 '16 at 4:40
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    $\begingroup$ I don't think that quantity is conserved, it is my understanding that it should be a non-increasing quantity. $\endgroup$ – nicoguaro Sep 15 '16 at 14:17

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