= Variable-coefficient conservative diffusion equation
{title2=$u_t=(a(x)u_x)_x$}
With positive diffusion coefficient $a(x)$, this <diffusion equation> expresses conservation with flux $-a(x)u_x$. For a differentiable coefficient, its spatial operator is $a u_{xx}+a'u_x$, rather than either $a u_x$ or $a u_{xx}$ alone. A <symmetric half-grid diffusion consistency> discretization uses the difference of the two adjacent interface fluxes, preserving that divergence structure. Suitable <Dirichlet boundary conditions> make the spatial operator dissipative in the <L2 norm>.
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