# Problem with solving coupled ODE and DAE equations with mass matrix (Error using daeic12 (line 77) This DAE appears to be of index greater than 1)

I am trying to solve 6 ODE equations coupled with 1 DAE one. The ODE equations have been discritized in space domain and ode15s MATLAB solver is used to solve the equations in time domain. I have defined a diagonal mass matrix including unity elements for ODEs and zero values for DAE equation. Assuming that "n" is the No. of nodes along the space, M has been defined as follows:

  O=ones(1,6*n);
Ze=zeros(1,n);
dia=horzcat(O,Ze);
M=diag(dia);


Here is the important parts of my main and function m.files:

.
.
.

y0(1:n)=zeros(n,1); %C0
y0(n+1:2*n)=zeros(n,1); %qc0
y0(2*n+1:3*n)=Tini*ones(n,1); %T0
y0(3*n+1:4*n)=Tini*ones(n,1); %Ts0
y0(4*n+1:5*n)=zeros(n,1); %N0
y0(5*n+1:6*n)=zeros(n,1); %qN0
y0(6*n+1:7*n)=zeros(n,1); %U0
t_int=1;
tf=3000;
nt=(tf/t_int);
tspan=(1:t_int:tf);
options=odeset('Mass',M);


and


.
.
.
dydt=zeros(7*n,1);
y(1,1)=Cin(1); %BC of C
y(n,1)=y(n-1,1); %BC of C
y(2*n+1,1)=Tg; %BC of Tg
y(3*n,1)=y(3*n-1,1); %BC of Tg
y(4*n+1,1)=Cin(2); %BC of N
y(5*n,1)=y(5*n-1,1); %BC of N
y(6*n+1,1)=Q/A; %BC of U
C(1:n,1)=y(1:n,1);
qc(1:n,1)=y(n+1:2*n,1);
T(1:n,1)=y(2*n+1:3*n,1);
Ts(1:n,1)=y(3*n+1:4*n,1);
N(1:n,1)=y(4*n+1:5*n,1);
qN(1:n,1)=y(5*n+1:6*n,1);
U(1:n,1)=y(6*n+1:7*n,1);

for i=2:n-1

dqcdt(i,1)=K1*(qsc-qc(i));
dCdt(i,1)=aEC(i).*C(i+1)+aWC(i).*C(i-1)-aPC(i).*C(i)+dqcdt(i-1));
dqNdt(i,1)=K2*(qcn-qN(i));
dNdt(i,1)=aEC(i).*N(i+1)+aWC(i).*N(i-1)-aPC(i).*N(i))+dqNdt(i-1);
dTdt(i,1)=aET(i).*T(i+1)+aWT(i).*T(i-1)-aPT(i)+*T(i))+Ts(i);
dTsdt(i,1)=T(i)-Ts(i)+dqcdt(i)+dqNdt(i);
dUdt(i,1)=U(i)-U(i-1)+dqcdt(i)+dqNdt(i);

end
dydt(1:7*n,1)=[dCdt;0 ;dqcdt;0 ;dTdt;0 ;dTsdt;0 ;dNdt;0 ;dqNdt;0 ;dUdt;0];


The equation located at the last line (i.e. dUdt(i,1)=...) is the DAE equation which is recognized by the defined mass matrix (please see "M" parameter defined before). I am not sure if this method is correct to specify the DAE equations! Anyway,when I run these codes, I face the following error:

Error using daeic12 (line 77) This DAE appears to be of index greater than 1.

I checked the Jacobian and found that it is a n*n matrix at t0. There are two columns (the first and last ones) which are all zero and I think it makes the Jacobian matrix singular (as explained in https://ch.mathworks.com/help/matlab/math/solve-differential-algebraic-equations-daes.html).

Could you please guide me how to overcome this error and fix my code? Any help is appreciated.

Best regards

• The equations you have are just too complicated to figure out what is happening. Try to simplify things, for example by simplifying the right hand sides, reducing the number of equations, etc. – Wolfgang Bangerth Dec 27 '20 at 22:19
• Hi Wolfgang, Thank you for your reply. To be more clear, I removed all constant parameters from right hand side of the coupled equations. – Sara.M Dec 28 '20 at 7:11
• I agree that this problem is far too complicated to debug from looking at your code snippets. I suggest posting your original system of PDE in mathematical form. – Bill Greene Dec 28 '20 at 13:03
• Hi Bill. Thank you for your reply. I have uploaded my codes and equations at: mathworks.com/matlabcentral/answers/…. I really appreciate it if you could please take a look at my documents and let me know what problem causes this error. As I mentioned, all equations are ode except for the one containing "U". It also should be mentioned that the equations have been discretized using FVM. Thank you. – Sara.M Dec 28 '20 at 22:35
• Dear expers, please help me. It is a long time that I have gotten stuck on this error. Thank you. – Sara.M Dec 29 '20 at 21:26