System of PDEs which is tricky for PDEPE

I have a system of PDEs, mainly diffusion equations of the form:

T_t-(k(x)(T_x)_x=a*E^2

c_t-(D(x)*c_x)_x=d*(E_x+c_x-T_x)

(epsilon*E)_x=-b*c

Where a and d are constants and _t,_x represent partial differentiation w.r.t. t and x respectively. In terms of pdepe, I would have: f=0 for the third equation. Would this cause errors in the code?

I'm also struggling to see how I can input the boundary conditions.

Any suggestions?

[Bill Greene]

Can you provide more details about the initial and boundary conditions for the three dependent variables?

[Matthew Hunt]

T is fixed on the end boundaries. dc/dx=0 on both boundaries and E(t,0)=I, and E(t,1)=-I

[Torsten]

A boundary condition for E can only be given in one end point of the interval, thus at x=0 or x=1.

[Matthew Hunt]

That is the boundary condition, you can ignore one of them.

[Bill Greene]

Do you mean T=0 at the boundaries? What about the initial conditions on the three variables? At t=0, the boundary and initial conditions must be consistent and also satisfy your differential equations.

Answer 1

[Torsten]

"pdepe" is designed to solve systems of parabolic-elliptic pdes. Your third pde is hyperbolic in nature. Thus "pdepe" is not suited to solve your system.

You will have to discretize your equations in space and solve the resulting system of ordinary differential equations using ODE15S.

Look up "method-of-lines" for more details.

Best wishes

Torsten.

[Matthew Hunt]

Cheers,

I had a feeling that this might be the case. You will note however that it is possible to solve the hyperbolic equation completely thus giving E as a function of c essentially. Can I drop that equation somehow and use the integral of c as part of the code for is it too hard to do that?

[Torsten]

Since "pdefun" is called only for a single value of x, you don't have the complete vector for c available. Thus no chance to calculate E, I guess.

Answer 2

[Torsten]

You could try to use "pdepe" with the third equation differentiated:

(epsilon*E)_xx=-b*c_x

with Dirichlet boundary condition

E = E0

at one end of the interval and

(epsilon*E)_x + b*c=0

at the opposite end.

Best wishes

Torsten.

[Matthew Hunt]

E is an electric field, there is no magnetic field, so there exists a potential for it E=phi_x, say. However, the derivative of E also appears in the diffusion equation for c, and this will entail a new form of "f" in the pdepe routine which does not fit in with the boundary conditions for c.

I was thinking about using a crank-Nicholson scheme for these equations.

[Torsten]

But the E_x and T_x terms in the diffusion equation for c will be put in the source term s for this equation, not in the f-term.

[Matthew Hunt]

That still won't work as it won't fit the boundary equation for E.

Answer 3

[Matthew Hunt]

So I managed a work around to get pdepe to work with my system and the code I'm using is:

%This is the simplistic model of a lithium ion battery model which I
%cobbled together knowing very little about batteries. The constants and so
%forth come from experiment.
%Define some global constants and parameters:
global rho; rho=1;global c_th; c_th=1;
global a_1; a_1=1;global a_2; a_2=1;global b; b=1; global c_1; c_1=1;global c_2; c_2=1;
global I_app; I_app=1.5;
global k;global D; global epsilon;
k=1; %Thermal conductivity, most likely a function.
epsilon=1; %Electrical conductivity, most likely a function
D=1; %lithium ion diffusion, most likely a function
m=0;
x=linspace(0,1,100);
t=linspace(0,1,50);
%Initial conditions;
global T_0; global c_0; global phi_0;
T_0=x./x;
c_0=smooth_step(x,0,1/3);
E_0=I_app/(epsilon)-(b/epsilon)*cumtrapz(x,c_0);
phi_0=cumtrapz(x,E_0);

sol = pdepe(m,@battery_GE,@battery_ic,@battery_bc,x,t);
T = sol(:,:,1);
E = sol(:,:,2);
c = sol(:,:,3);

figure;
surf(x,t,T);
title('Temperature');
xlabel('Distance x');
ylabel('Time t');

figure;
surf(x,t,E);
title('Electric Field');
xlabel('Distance x');
ylabel('Time t');

figure;
surf(x,t,c);
title('Li ion concentration');
xlabel('Distance x');
ylabel('Time t');

function [CC, FF, SS]=battery_GE(x,t,u,DuDx)

CC=[rho*c_th; 0; 1];
FF=[a_1*k; epsilon; c_1*D].*DuDx;
SS=[a_2*sigma*DuDx(2); -b*u(3);c_2*(DuDx(3)-(b*u(3)/epsilon)-DuDx(1))];

function u0=battery_ic(x)
u0=[T_0;phi_0;c_0];

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
T_L=1;
T_R=2;
pl=[T_L; I_app; 0];
ql=[0; sigma/epsilon 1];
pr=[T_R; -I_app; 0];
qr=[0; sigma/epsilon 1];

I get the following error, which I don't understand because I made T_0 a global variable:

Undefined function or variable 'T_0'.

Error in battery_model>battery_ic (line 54)
u0=[T_0;phi_0;c_0];

Error in pdepe (line 229)
temp = feval(ic,xmesh(1),varargin{:});

Error in battery_model (line 24)
sol = pdepe(m,@battery_GE,@battery_ic,@battery_bc,x,t);

I'm not sure what is going on now.

[Bill Greene]

What is the definition of smooth_step?

[Matthew Hunt]

function y=smooth_step(x,a,b)
%This is a smooth approximation of the Step function done via the erf
%function
z=zeros(1,length(x));
if (a==x(1))
    m=find(x>b,1,'first');
    z(1:m-1)=1;
    y=min(z,0.5*(1+erf(-30*(x-b))));
elseif(b==x(end))
    n=find(x<=a,1,'last');
    z(n+1:end)=1;
    y=min(z,0.5*(1+erf(30*(x-a))));
else
    y=0.5*min(1+erf(30*(x-a)),1+erf(-30*(x-b)));
end

Answer 4

[Precise Simulation]

Systems of PDEs might also be easier to solve with the FEATool FEM Toolbox which features a GUI and easy syntax for defining custom PDEs and equations. This example of heat transport and diffusion might be a good start, otherwise you could perhaps use the built-in convection and diffusion physics mode and modify it according to your equations.

Answer 5

[Matthew Hunt]

I have some values which allow the code to run BUT I have oscillations in the diffusion which I should not get. I am assuming that this is a numerical instability, is it caused by too large a timestep?