Finite difference method to solve first-order, multivariable

[idraftscience]

I'm trying to replicate the model presented in this [paper](http://www.sciencedirect.com/science/article/pii/S1359431103000474), which is basically to model heat and mass transfer along a one-dimensional duct.

There are four characteristic equations for this problem :

Momentum conservation

$$\frac{\partial Y}{\partial t} +V\frac{\partial Y}{\partial z}+\omega_1\frac{\partial W}{\partial t}=0 $$

Mass transfer:

$$ \frac{\partial W}{\partial t} + \omega_2 (Y_w - Y) = 0 $$

Energy conservation:

$$ \frac{\partial T}{\partial t} + V\frac{\partial T}{\partial z} + \omega_3\frac{\partial T^*}{\partial t} = \omega_4 (Y-Y_w) $$and Heat Transfer

$$\frac{\partial T^*}{\partial t} + \omega_5 (T^* - T) + \omega_6 (Y_w - Y) + \omega_7 (Y_w - Y)(T-T^*) = 0$$

I'm attempting to use a forward-difference approximation method, i.e.
$$ \frac{\partial Y}{\partial z} = \frac{ Y_{i+1}^m - Y_i^m }{\Delta z}$$
and so on for the other variables, where i is the spatial index in z and m is the time index. The $\omega$s and $V$ are constants.

I have initial and boundary values for all the parameters, so that leaves me needing to find $$Y_i^{m+1}, Y_{i+1}^m, W_i^{m+1}, T_i^{m+1}, T_{i+1}^m, T_i^{*m+1} $$

I have four equations and six unknowns. The paper states that an 'implicit up-wind difference form' and Gauss-Jordan elimination is used, but I can't see how my approach differs from that. I do acknowledge that my approach is an explicit method, but I believe the issue of indeterminacy remains. Am I missing something in my approach?

 

[dawin]

What are the boundary conditions for this particular problem, and why are you trying to use an explicit method? Is there a particular reason?

I don't see your explicit formulation, only a first-order approximation to the partial derivative.

Depending what information they give you, I'd think that the implicit method is the easier method to use in this case. I'm interested in what the assumptions/BCs are though.

 

[idraftscience]

Initial conditions given are T(z,0), Y(z,0), T*(z,0), W(z,0), Yw(z,0), and boundary conditions are T(0,t) and Y(0,t). My formulation is to replace the approximation into the four equations above, and separate the parameters such that they can be represented in the form Ax=b, where x are the parameters T,Y,W, etc. So I guess I'm not sure whether that approach is explicit/implicit.

I'm trying to verify that my approach is valid, since it seems to be in order with other examples I've seen of first-order pde methods.

 

[pasmith]

I have initial and boundary values for all the parameters, so that leaves me needing to find $$Y_i^{m+1}, Y_{i+1}^m, W_i^{m+1}, T_i^{m+1}, T_{i+1}^m, T_i^{*m+1} $$

I have four equations and six unknowns.

$Y_{i+1}^m$ and $T_{i+1}^m$ are known: you were either given them as boundary or initial conditions, or you calculated them from the values at the previous time step.