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Let's take the example of the unsteady one-dimensional heat equation inside a solid on a domain $x\in[0,1]$: $$\partial_t u - D\partial_{xx} u = 0$$ with the initial profil $u(0,x) = u_0(x)$ at $t=0$, and Dirchlet boundary conditions enforcing that the wall at $x=0$ (respectively $x=1$) is at temperature $u_{L}$ (respectively $u_{R}$). If we use discretise ...