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2

Mass is conserved always. That's the known fact. To show this holds true in convection-diffusion equation, I need to introduce material derivative to you. The material derivative of a scalar quantity as $C(\mathbf{r},t)$ is defined as: $$\frac{D C(\mathbf{r},t)}{D t} = \frac{\partial C}{\partial t} + \mathbf{v} \cdot \nabla C$$ Where $\mathbf{v}$ is the ...

6

The short answer Eight order means it approximates $u$ with nine points and the local truncation error will be $O(\Delta{}x^8)$. Higher the order, better the approximation. The eight order mentionned is the accuracy order of the finite difference, nothing to do with the derivation order. Have a look at this. The long answer Let us note $n$ the order of ...

1

In essence, you are asking whether you have values for a function $h'(r)$ at points $r_i$, you can obtain an approximation of $h(r)$. The answer is of course yes: If you connect the points $(r_i,h'(r_i))$ by a piecewise linear curve, then you can integrate that to obtain a piecewise quadratic approximation of $h(r)$. You can be more accurate if you connect ...

4

The condition that has to be satisfied at the node point 2 is that the influx equals the outflux. Here, this will then be $$v_{12}C_{12} = v_{24}C_{24}+v_{23}C_{23},$$ but $C_{ij}$ is not the average concentration in each pipe: It is the concentration at end point 2 of the pipes. If the velocities are constant in time, then you can of course rewrite ...

2

We've been going back and forth in the comments and thanks for adding so much more information. So the first thing I'm noticing is that if you have many variables, but they depend on each other through equations of state and other equations they are therefore coupled. I think it's best to interpolate the minimum number of variables required (maybe ...

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