I am finding issues in PDEPe which shows time integration failure again and again

"Warning: Failure at t=2.088125e-04. Unable to meet integration tolerances without reducing the step size

below the smallest value allowed (4.336809e-19) at time t. "

This message is being shown for running my code. I am uploading my code below.

clear all

clc

close all

m=0;

x=linspace(0,1,1001);

t=linspace(0,1,1001);

options = odeset('RelTol', 1e-3, 'AbsTol', 1e-6,'MaxStep', 0.1);

sol=pdepe(m,@coupledpde_BACD,@coupledic_BACD,@coupledbc_BACD,x,t,options);

b=sol(:,:,1);

figure(1)

surf(x,t,b)

xlabel('x')

ylabel('t')

zlabel('b(x,t)')

grid on

title('Bacterial distribution');

view([150 25])

colorbar

colormap jet

for j=[1 11 101]

ylim([0,1.2])

figure(2)

hold on

plot(x,b(j,:),LineWidth=2)

title("Particle Concentration Profile at t= (1, 11, 101)*td")

xlabel("Characteristic length")

ylabel('bacteria Concentration')

end

function [c,f,s]= coupledpde_BACD(x,t,u, dudx)

Db=10^-12;

Dc=10^-9;

X0=10^-9;

% X0=linspace(10^-11,10^-9,100);

% Db=linspace(10^-13,10^-12,100);

c=1;

s=0;

l=[pi/2 3*pi/2 5*pi/2];

A=4.*(cos(l)-1)./(2.*l-sin(2.*l));

mob_ratio=X0/Dc;

d_ratio=Db/Dc;

f=(d_ratio).*dudx-u*(mob_ratio)./(1+exp(-l(1)^2.*t).*A(1).*sin(l(1).*x)).*(exp(-l(1)^2.*t).*A(1).*cos(l(1).*x).*l(1));

end

function u0 = coupledic_BACD(x)

u0=1;

end

function [pl,ql,pr,qr]= coupledbc_BACD(xl,ul,xr,ur,t)

pl=ul;

ql=0;

pr=0;

qr=1;

end

please help me find the issue in the code

[Torsten]

Which boundary condition do you try to set at the right end of your spatial interval ?

[Aadityaamlan]

Zero neumann boundaty condition

[Torsten]

At the moment, you set

(d_ratio).*dudx(@x=1)-u(@x=1)*(mob_ratio)./(1+exp(-l(1)^2.*t).*A(1).*sin(l(1))).*(exp(-l(1)^2.*t).*A(1).*cos(l(1)).*l(1)) = 0

Is this what you mean by "zero neumann boundary condition" ?

[Aadityaamlan]

This is the equation itself

Answer 1

[Umang Pandey]

Hi Aaditya,

This error message indicates that the problem you are trying to solve is too stiff for the ODE solver you are trying to use, or that it includes a singularity.

It has been explained by the MathWorks Support Staff in the following MATLAB answer:

https://www.mathworks.com/matlabcentral/answers/101581-why-do-i-receive-a-warning-about-integration-tolerance-when-using-the-ode-solver-functions#answer_110929

Hope this helps!

Best,

Umang