Heat equation, discretization, stability condition, finite differences, temperature propagation, spatial discretization interval, time step
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This document discusses the discretization of the heat equation and the choice of time step and spatial discretization interval for stable temperature propagation.
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[...] Writing a Python program to solve a simple problem Discretization of the problem: On the heat equation as follows: . We use finite differences to discretize this equation. One notes the spatial discretization interval, and the time step. On utilise l'indice for the position and for the time. On this way who becomes et becomes . So we have: D'ou . [...]
[...] This last equation is used in the code to propagate the temperature over time throughout the entire thickness of the studied Earth's crust. Choice of time step and spatial discretization interval: To have sufficiently 'smooth' plots, we need a large number of points on the position axis . One will then take for example m. For the choice of time step, we first take day. We observe that the heat equation propagation is unstable, the temperature diverges. We then reduce the time step (by dividing by 10) until the system becomes stable. [...]
[...] We then find day. Editor's Note : I don't know if you saw this point in class, but we actually have to respect the following stability condition: . So for and one obtains what is well below [...]
