How to solve partial differential algebraic equations

New to Julia
1

Hi all,

I am quite new to Julia and programming in general. I am currently doing a modelling PhD and I have a set of couple Partial Differential Algebraic Equations (PDAE). I tried solving them on comsol but it is extremely cumbersome trying to rewrite the equations into a format that comsol accepts. So I was wondering if there is a PDAE solver in Julia since I want to eventually shift to a Julia solver for my PhD. Any help will be appreciated thanks in advance.

1 Like
2

Here is a guide to the pde ecosystem in Julia: GitHub - JuliaPDE/SurveyofPDEPackages: Survey of the packages of the Julia ecosystem for solving partial differential equations

EconPDEs.jl for example solves pdaes, but a very specific type that arise in economics (from HJBs with nonlinear concave objective functions)…
I don’t think there is a general package yet

3

what kind of PDE do you have?

4

Hi Chris,

Thank you for replying.

WhatsApp Image 2024-05-08 at 17.41.06_340e01be
WhatsApp Image 2024-05-08 at 17.41.06_340e01be1131×1599 101 KB

The equations are a set of equations for modelling a battery but unfortunately I cannot use existing packages such as JuBatt since the configuration is very different than them. The solver needs to solve for 7 dependent variables - 5 concentration terms C_OH, C_Zn, C_ZnO, C_H20, C_K and two potential variables phi_l and phi_s. So as you can see the equation 5 is the problematic term here since it is an algebraic equation so I am not sure how to write it in a PDE solver in Julia.

Also on top of that, the exponent adds non linearity to the equations so I am not to able to right away start solving this by direct discretization.

PS Apologies for the handwritten equations. Please let me know if you do not understand something in there.

5

What assumptions are you making about this system? Is this within a single section of the cell, such as the electrode? Are the diffusivities and conductivities constant? I would check out MethodOfLines.jl for simulating coupled PDAEs. It handles the discretizations, boundary conditions, algebraic equations, and fully supports nonlinear terms such as the BV kinetics.

1 Like
6

Thank you for your reply @MarcBerliner. I will check MOL package first to look at the possibility.
The diffusivities and conductivities are actually functions of concentration but now for the sake of simplicity to get started i am assuming them to be constant. The model is for a spherical coordinate system.

7

@Anish_Kumar , I also solve similar equations for my research. I can see that you are solving, non-linear convection-diffusion-reaction problem. The usual procedure for solving such problems is this:

  1. convert the differential-equations (time-independent- in your case i think it is 5,6 and 7) to difference equations. MethodOfLines.jl must surely be of help. Or if you are good at FEA coding, you can construct and load the sparse-stiffness matrix into julia. Even if youre good at FEA, its a good idea to use julia inbuit poisson solvers in case youre working with higher dimensions 3D problem.
  2. I can see that you have 5 species, give appropriate initial conditions, and use [differentialEquations.jl] package to solve the ODEs. In case you are using your own FEA routine, In the ODE function definition -remember to include solution to potential (Phi = InvK*F, ) using this command:
    prob = LinearProblem(K_stiffness , Force);
    @time sol = solve(prob,LinearSolve.KrylovJL_CG(),maxiter = 1000);
    Code the other dynamics appropriatey, look into differentialEquations.jl documentation for the same.
1 Like