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I am trying to solve the diffusion equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ via finite volume method and time-stepping via implicit methods and so I essentially have a discretization that looks like:

$$ \frac{\bar{u}^{n+1}_{i} - \bar{u}^{n}_{i}}{\Delta t} = \frac{u^{n+1}_{i+1} - 2 u^{n+1}_{i} + u^{n+1}_{i-1}}{(\Delta x)^2} $$

However, this creates a problem because for N nodes you have N equations, but because of the $\bar{u_i}$ you gain an additional N variables, leaving you with N equations and 2N variables.

I'm sure that finite volume PDEs are solved every day using implicit methods so there must be something easy that I'm missing but I can't seem to find anything anywhere. How would I go about solving this problem?

  • 0
    What are $\bar{u}^{k}$? – 2017-02-27
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    @VorKir the. Exact value – 2017-10-12

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I assume that $\bar{u}$ must be defined from other values on the grid (i.e. sum average of surrounding values at a point in time). Then, you do not get $2N$ additional values because each of $\bar{u}$. Anyhow, you cannot have $2N$ variables for $N$ equations and expect a unique solution.

Assuming I am right, the solution to your system of equations (SoE) still demands that you solve an implicit system. Here, it is optimal (I think) to use the Newton-Raphson method to find the solution at every time step. Since your SoE is linear you can obtain the Jacobian exactly by finite differences (this is the only computing intensive part). The Newton-Raphson will then require a single iteration.