Set up a tridiagonal for a system of equations

[Hypatio]

I am trying to solve the system of simultaneous equations:
$$\frac{\partial \rho_f\phi}{\partial t}+\frac{\partial}{\partial z}(\rho_f\phi v_f)= \frac{\partial F}{\partial t}$$
$$\frac{\partial \rho_s(1-\phi)}{\partial t}+\frac{\partial}{\partial z}(\rho_s(1-\phi) v_s)=-\frac{\partial F}{\partial t}$$
and
$$v_f-v_s=-\frac{k}{\phi \mu}\frac{\partial P}{\partial z}$$
where v_s and v_f are unknown except v_s is known at the lower boundary in z.
I will use explicit finite differences such that, I think, the discretized forms are
$$\rho_f^n\phi^n-\rho_f^0\phi^0+\frac{\Delta t}{\Delta z}[\rho_f^n\phi^nv_f^n-\rho_f^0\phi^0v_f^0]=F^n-F^0$$
$$\rho_s^n(1-\phi^n)-\rho_s^0(1-\phi^0)+\frac{\Delta t}{\Delta z}[\rho_s^n(1-\phi^n)v_s^n-\rho_s^0(1-\phi^0)v_s^0]=-(F^n-F^0)$$
and
$$v_f-v_s=-\frac{k}{\phi \mu}\frac{P_{z+1}-P_z}{\Delta z}$$
where superscripts n and 0 are for new time and current time, respectively.
But I do not understand what the strategy should be for building a tridiagonal matrix with these equations. What should I be doing? How can I set this up?

More on terms: v_f and v_s are velocities, rho_f and rho_s are densities, \phi is volume of material moving at velocity v_f, F is a mass flux, and k and
\mu are constants.

 

[jvicens]

Thank you for sharing your system of equations. It seems like you are working on a transport problem involving two different materials (represented by the subscripts f and s). In order to build a tridiagonal matrix for this system, you will need to first discretize the equations using the explicit finite difference method as you have done. This will result in a set of equations for each time step and grid point.

Next, you will need to rearrange the equations in order to have all the unknowns on one side and known terms on the other side. This will allow you to create a matrix equation of the form Ax=b, where A is a tridiagonal matrix, x is a vector containing the unknown variables, and b is a vector containing the known terms.

To create the tridiagonal matrix, you will need to consider the discretized equations for each grid point and time step. Each equation will contribute three terms to the diagonal, upper diagonal, and lower diagonal of the matrix. The exact values of these terms will depend on the specific discretization scheme you are using.

Once you have the matrix A, you can solve for the unknown variables using a tridiagonal solver, such as the Thomas algorithm. This will give you the values of v_f, v_s, and \phi at each grid point and time step.

I hope this helps guide you in building the tridiagonal matrix for your system of equations. If you have any further questions, please don't hesitate to ask. Good luck with your research.