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Consider a nonlinear advection-diffusion equation of the form

$$ \frac{\partial u}{\partial t} = \nabla \cdot (a(u) \nabla b(u) - \vec{c}(u)u) \tag{1} $$

on a rectangular domain with Dirichlet boundary conditions. We have $a(u) > 0$ for all $u \in \mathbb{R}$ and $b(u)$ is a monotonically increasing function of $u$. I wish to solve (1) by finite differences.

Can (1) always be discretized in such a way such that all matrix solves involve only symmetric positive definite matrices? If not, are there conditions on $a$, $b$, and $\vec{c}$ such that this is the case?

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  • $\begingroup$ Can you specify your desired temporal discretization? $\endgroup$ – Charles Jun 11 '17 at 2:22
  • $\begingroup$ @Charlie Either Crank Nicholson or Backward Euler $\endgroup$ – eepperly16 Jun 11 '17 at 2:24
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One strategy may be to use a semi-implicit temporal discretion, treating the diffusive term implicitly and the advective term explicitly. Doing so alleviates the extreme timestep restriction typically imposed by explicit diffusion, and it keeps your linear system symmetric. This is a common trick with (eg) the Navier Stokes equations to avoid precisely the same issues.

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