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I am attempting to solve a nonlinear advection diffusion equation

$$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x} + u^2)$$

with Robin boundary conditions via the finite volume method.

I am using the WENO method to reconstruct the function $u$ and its derivative $u_x$ as well as the numerical flux $\hat{f}$ all from the average values $\bar{u}$ and so my discretization currently looks like: $$ \frac{\bar{u}^{n+1} - \bar{u}^{n}}{h} = \hat{f}_{i+\frac{1}{2}} - \hat{f}_{i-\frac{1}{2}} $$

My question is:

If I use an implicit time discretization, do I need to use Newton's method (or any other known method to solve nonlinear equations) to advance a time step?

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    $\begingroup$ What time levels are your $\hat{f}$ defined on? If time step $n$ then it's explicit and you don't need e.g. Newton's method, if depends on $n+1$ then it is also an unknown variable. $\endgroup$ – Steve Feb 22 '17 at 10:02
  • $\begingroup$ In terms of searching for literature, it might be useful to note that with a change of variables $\bar{x}:=-2x$ you have viscid Burgers' equation. $\endgroup$ – Steve Feb 22 '17 at 10:25
  • $\begingroup$ I would like to use an implicit method, so my $\hat{f}$ depends on $n+1$. $\endgroup$ – hijasonno Feb 22 '17 at 17:17
  • $\begingroup$ If you're using WENO and an implicit time stepping method, in general you have to solve a nonlinear system of equations at each step. It doesn't matter what your PDE is. $\endgroup$ – David Ketcheson Feb 22 '17 at 23:42
  • $\begingroup$ I've edited the question to reflect what it seems you really want to ask. Feel free to edit it more if I've misunderstood. $\endgroup$ – David Ketcheson Feb 22 '17 at 23:44
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Generically, yes. Since you have a nonlinear PDE you will end up with a nonlinear algebraic system no matter what spatial discretization you use. With WENO you will have a more strongly nonlinear system and (at high Reynolds number) it will be hard to find a convergent solver.

A common approach for this kind of equation is to use an IMplicit-EXplicit (IMEX) additive Runge-Kutta method, in which the explicit part is applied to the nonlinear convection term (avoiding the need to solve a nonlinear algebraic system) and the implicit part is applied to the linear diffusion term (avoiding the need for an excessively small timestep).

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