Need help solving PDE, numerically which contains some transformations

[re444]

Hi, hope this is a right place to ask this question. I work in the soil physics field and this problem has taken lots of my energy for a while! let's state it:

Unsaturated horizontal water flow in 2 layer soil:

we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic head), x , t (for time).

M(x,t,h) , K(x,t,h) , h(x,t).

PDE is of the form: ∂M/∂t = ∂/∂x (K ∂h/∂x ) .

on the domain (x=[0,xL] ) is divided into 2 layers (x1 & x2) each with its own M and K (2 soil type).
So I have M1, K1 for first layer and M2 & K2 for the second. Instead of solving these two PDE, by applying some linear transformations of the kind:

M*= k1.M ; K*= k2.K ; h*= k3.h ; x1*=k4.x1 ; x2*= k5.x2 ; t*= t .

I'm solving one unique PDE* in the domain (x1* $\cup$ x2*) with transformed boundary(at the upper of 1st layer and lower of 2nd layer) and initial conditions. Now here is my problem: when I get PDE* solutions and then transforms theme back, I get answers different from when not applying this whole transformation procedure.

I guess the question is: when I do a series of [linear] transformations to a PDE is it true to transform the solution back using the inverse of that transformations?

tried my best to express the problem,
thanks

 

[Chestermiller]

See Freeze and Cherry, "Groundwater."

 

[re444]

See Freeze and Cherry, "Groundwater."

I want to look at it from Differential Equations point of view. Just consider the last question at the bottom of 1st post.

 

[Chestermiller]

Let me see if I understand correctly. When you say that the flow is unsaturated, you mean that, within your two soil layers, there is an interface between the water below and the air above, and that this interface is moving as a function of time. Correct?

Chet

 

[blue_raver22]

Dear researcher,

Thank you for reaching out with your question. It sounds like you are working on a challenging problem in soil physics. Solving PDEs numerically can be a complex task, especially when transformations are involved.

To answer your question, yes, it is generally true that if you apply a series of linear transformations to a PDE, you should be able to transform the solution back using the inverse of those transformations. However, there are a few potential issues that could be causing the discrepancies you are seeing.

First, it is possible that there is an error in your transformation procedure. I would suggest double-checking your calculations and making sure that the transformations are being applied correctly.

Second, it is possible that the transformations are introducing numerical errors into your solution. This can happen if the transformations involve very large or small numbers, which can lead to rounding errors. If this is the case, you may need to use a more precise numerical method or adjust your transformations to avoid these errors.

Lastly, it is important to consider the validity of the transformations themselves. Are the transformations you are applying appropriate for the problem you are trying to solve? Are they physically meaningful in the context of soil physics? If not, this could be the source of the discrepancies you are seeing.

I hope this helps you in your research. If you continue to have problems, I would suggest consulting with a colleague or mentor who may have more experience with PDEs and transformations in your specific field. Best of luck with your work!