# Can this nonlinear advection-diffusion equation be discretized as to only have to solve SPD systems?

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?

• Can you specify your desired temporal discretization? – Charles Jun 11 '17 at 2:22
• @Charlie Either Crank Nicholson or Backward Euler – eepperly16 Jun 11 '17 at 2:24