(How) Can I (numerically) integrate u from z=0 to z=z in a=acoeffun​ction(loca​tion,state​), when using PDE toolbox and a PDEModel for general PDE?

I am in the process of figuring out if I can use either pdepe.m or the Partial Differential Equation Toolbox to solve a set of two partial differential equations,

and ,

where and .

The problem I'm having, is that the following integral needs to be evaluated in both and :

at each time-step and for each depth z in the "depth-mesh".

As far as I can tell, using pdepe.m is not feasible, since the input u in pdefun is a Nx1 vector, where N is the number of equations in your system.

I was hoping that using the Partial Differential Equation Toolbox, eventhough it's intended use is for PDEs with two or more spatial variables, would allow me to integrate numerically from z=0 to z= current depth.

Answer 1

[SAI SRUJAN]

Hi Anita,

I understand that you are trying to integrate'u' from 'z=0' to 'z=z'.

Given that you are solving a set of two partial differential equations and need to perform an integral at each time step and for each depth ('z') in the depth mesh, here's a general approach to accomplish this:

% define model
model = createpde();
% specify PDE coefficients
specifyCoefficients(model, 'm',0, 'd',0, 'c',c_coeff, 'a',a_coeff, 'f',f_func);
% solve the PDE for the time span of interest.
result = solvepde(model, tlist);

% Assume 'z_values' are the z-coordinates and 'u_values' are the solution values
z_values = ...; % Extract from the mesh
u_values = ...; % Extract from 'result'

% Integrate from z=0 to z=z_current
z_current_index = find(z_values <= z_current, 1, 'last');
integral_value = trapz(z_values(1:z_current_index), u_values(1:z_current_index));

For numerical integration, MATLAB's 'trapz' function or the 'integral' function can be used. Please refer to the following documentation to understand more about the forementioned MATLAB functions.

https://www.mathworks.com/help/matlab/ref/trapz.html

https://www.mathworks.com/help/matlab/ref/integral.html

I hope this helps!

[Anita Løtvedt]

Thank you for taking the time to help!

The issue is that these integrals are a part of the right hand side of the equations, such that they must be evaluated in each time step. While this can easily be taken care of by writing my own solver, or simply semi-discretizing the system of PDEs and using one of the ODE solvers, I was hoping that there was some neat trick/function in MATLAB or the PDE toolbox that would save me some time.

[Torsten]

If an answer is still of interest after 2 1/2 years:

You can use

https://uk.mathworks.com/matlabcentral/fileexchange/97437-pde1dm

This code is a variant of MATLAB's "pdepe", but the vector u is supplied in all grid points x for each time t. This makes it possible to use MATLAB's "trapz" to evaluate the integral.

[Anita Løtvedt]

It is, thank you! I will take the code into consideration.