solving PDE for convection diffusion

[ymharshe]

I am solving classical convection-diffusion problem which takes the following form.

with B.C.

for the moment I am taking constant D and U

My code looks like as follows.

from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

diffusivity = 1e-9
velocity = 1e-6
cmax = 0.6

term_1 = f"({diffusivity}) * laplace(c)"
term_2 = f"({velocity}) * gradient(c)"

bc_lower = {"type": "mixed", "value": -velocity/diffusivity, "const": 0}
bc_upper = {"type": "mixed", "value": -velocity/diffusivity, "const": 0}

eq = PDE({"c": f"{term_1} - {term_2}"}, bc=[bc_lower, bc_upper])

grid = CartesianGrid([[0, 0.08]], 8000)  # generate grid
field = ScalarField(grid, 0.2)  # generate initial condition


storage = MemoryStorage()  # store intermediate information of the simulation
res = eq.solve(field, 86400, dt=1e-1, tracker=storage.tracker(1))  # solve the PDE

plot_kymograph(storage)  # visualize the result in a space-time plot

Can anyone tell me if this implementation is correct or not? Also, I get some errors while running the code.

[ymharshe]

The code above, does not have the

term.
As I wanted to verify that the code works without this term. But then I would like to eventually have the code including this term as mentioned in the first post here.

Thanks in advance for your feedback and support.
Regards,
Yogesh

[ymharshe]

The boundary conditions is such that there is a term in the bracket which has a summation of all dependent variable (phi at each z). phi is matrix n entities in z spaces. Please see, if it can be handled by py-pde.

[david-zwicker]

It sounds as if you want to have an integral over the field in the boundary condition, which is currently not supported.

[ymharshe]

it is like system of PDEs solved simultaneously. Can py pde handle a set of PDEs solved simultaneously with different initial conditions? The boundary condition says that the space derivative of dependent variables of the system of PDEs at both boundaries follow this condition. The sum is the sum of dependent variables at the boundaries.

[david-zwicker]

py-pde supports systems of PDEs, but coupling these equations via boundary conditions is not supported out of the box. You could develop a work-around by using the advanced UserBC class, which allows you to set virtual points yourself. This requires quite detailed knowledge of the internal workings and I unfortunately don't have the bandwidth to help you with the implementation.

[ymharshe]

is there any example of solving system, of PDEs with individual initial conditions and boundary conditions? Actually, the BC I have is similar to the Robin condition where the space derivative is equal to some constant multiplied by the dependent variable and (1- sum of all dependent variable values at the boundary conditions/ constant)