How to numerically solve a system of coupled partial differential and algebraic equations?

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Arpita 2023-1-25

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回答： Sarthak 2023-3-9

I have a system of coupled partial differential and algebraic equations.

Two 1-D parabolic pdes coupled (function of x and time) with two algebraic equations. What would be the way to solve this sytem?

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Arpita 2023-2-2

编辑：Torsten 2023-2-2

在 MATLAB Online 中打开

Porada1D_2023_01_31.m

I don't think I have any programming errors, but could you possibly help me?

If the issue is the difficulty of the system to be integrated, should I move to a different software?

clear
clc
%% Parameters %%
% Experimental conditions %
p.Q1=37.33/60;% 4.4664/60; % flowrate to the stack (mL/s)
p.I=-1258/1120*10^(-3); % applied current density (A/cm2)
p.T1=500;% round(41/Q1); % charging time(s)
p.C0=0.0342; % initial concentration (mmol/ml)
% Electrode and cell physical properties %
p.Mass=10; %electrode weight (g)
p.N=1; % number of electrode pairs
p.psp=0.708; % spacer porosity
p.pma=0.4; % macroprosity
p.pmi=0.28;  % microporosity
p.Lelec=0.028; % each electrode thickness (cm)
p.Lsp=0.01;  % spacer thickness (cm)
p.a=1120; % each electrode area (cm^2)
p.hrt1=p.Lsp*p.a*p.N/p.Q1; % hydraulic retention time (s) in the spacer
p.tlag=round(1*p.hrt1);
p.alpha=50;
p.Rext=200;% resistance (om*cm2)
% Electrochemical properties %
p.De=1.68*10^(-5);  % effective diffusion coefficient (cm^2/s)
p.R=6.1;% specific electrode resistance (om*mmol/cm), from zhao, WR, 'optimation' 2013
p.u=0; % chemical attraction term (kT) 
p.Vt=0.0256; % thermal voltage (V)
p.Cst1=72; % stern capacitance (F/mL) value from JE.Dykstra W 2016.
p.F=96.5; % farady constant (C/mmol)
p.i=p.I/p.F; % convert (A/cm2) to (mmol/s/cm2)
p.Vapplied = (1.2/2/p.Vt);
p.V1 = p.Vapplied-(p.i*p.Lsp/(4*p.C0*p.psp*p.De));
% using method of lines %
tf      = 200;                               % End time
p.nz    = 100;                                  % Number of nodes
x       = linspace(0,p.Lelec,p.nz);                 % Space domain
dx      = diff(x);
p.dx = dx(1);                % Space increment
%% Initial conditions %%
Ceff_0 = p.C0*(p.pma+p.pmi); % Ceff(x,0)= C0
Cch_0 = 0; % Cch(x,0) = 0
Cma_0 = p.C0; % Cma(x,0) = C0
phima_0 = p.V1; %phima(x,0) = V1
Csp = p.C0.*ones(p.nz,1);
Ceff = ones(p.nz,1);
Ceff=Ceff_0.*Ceff;
Cch = zeros(p.nz,1);
Cma = ones(p.nz,1);
Cma=Cma_0.*Cma;
phima = ones(p.nz,1);
phima=phima_0.*phima;
%y0      = [Csp;Ceff;Cch;Cma;phima];
y0      = [Ceff;Cch;Cma;phima];
tspan   = [0 tf];
%% Boundary conditions %%
dCmadx = 0; % dCmadx(Lelec,t) = 0
dphimadx =0; % dphimadx(Lelec,t) =0
%% Mass matrix (for DAE system) %%
M = eye(4*p.nz);
for i=(2*p.nz+1):(4*p.nz)
    M(i,i)=0;
end
options = odeset('Mass',M,'RelTol',1e-3,'AbsTol',1e-10);%'MassSingular','yes');
[t,y] = ode23t(@(t,y)porada1D(t,y,p),tspan,y0,options);
%% Plot %%
Ceff_matrix=y(:,1:p.nz);
Cch_matrix=y(:,p.nz+1:2*p.nz);
Cma_matrix=y(:,2*p.nz+1:3*p.nz);
phima_matrix=y(:,3*p.nz+1:4*p.nz);
plot(t,Ceff_matrix(:,:))
grid
xlabel('time')
ylabel('Ceff')
%title('Dimensionless concentration at ''z = L''')
%figure()
%plot(t, phima_matrix(:,:))
%% function %%
function out = porada1D(~,y,p)
% Variables
Ceff = y(1:p.nz,1);
Cch = y(p.nz+1:(2*p.nz),1);
Cma = y((2*p.nz)+1:(3*p.nz),1);
phima = y((3*p.nz)+1:(4*p.nz),1);
%phima = y((4*p.nz)+1:(5*p.nz),1);
% Space derivatives
dCmadx = zeros(p.nz,1);
d2Cmadx2 = zeros(p.nz,1);
dphimadx = zeros(p.nz,1);
d2phimadx2 = zeros(p.nz,1);
dCmadx(2:end-1) = (Cma(3:end)-Cma(1:end-2))./(2.*p.dx);
d2Cmadx2(2:end-1) = (Cma(3:end)-(2*Cma(2:end-1))+Cma(1:end-2))./(p.dx^2);
dphimadx(2:end-1) = (phima(3:end)-phima(1:end-2))./(2.*p.dx);
d2phimadx2(2:end-1) = (phima(3:end)-(2*phima(2:end-1))+Cma(1:end-2))./(p.dx^2);
% Governing equations
%dCspdt = (2.*p.pma.*p.De/p.psp/p.Lsp).*dCmadx + ((1./p.psp./p.hrt1).*(p.C0-Csp));
dCeffdt = p.pma.*p.De.*d2Cmadx2;
dCchdt = 2.*(p.pma./p.pmi).*p.De.*(dCmadx.*dphimadx+Cma.*d2phimadx2);
phit = phima+(asinh(Cch./(-2.*Cma)))-(p.F.*Cch./(p.Vt.*(p.Cst1+p.alpha.*(Cch.*Cch))))-p.V1;
Cefft = Ceff-p.pma.*Cma-(p.pmi.*Cma.*cosh(Cch./(-2.*Cma)));
% Boundary conditions
Cma(1,1)=p.C0;
dCmadx(end)=0;
dphimadx(end)=0;
%out = [dCspdt;dCeffdt;dCchdt;phit;Cefft];
out = [dCeffdt;dCchdt;phit;Cefft];
end

Arpita 2023-2-3

The version I sent you had the boundary conditions after the governing equations. Sorry about that.

Even with boundary conditions before the governing equations, I see the same problem.

With all three boundary conditions, I get the error that the DAE is greater than 1.

When I only use the dCmadx = 0 and dphimadx = 0 as the boundary conditions, I get the error: " Unable to meet integration tolerances without reducing the step size below the smallest value allowed (4.336809e-19) at time t."

Torsten 2023-2-3

You should start with a simpler problem from which you know that potentially arising problems with the integrator stem from your programming, not from the difficulty or even unsolvability of the problem itself.

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Answer 1

Sarthak 2023-3-9

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此回答的直接链接

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Hi,

One way to solve a system of coupled partial differential equations (PDEs) and algebraic equations is to use a numerical method such as finite difference or finite element method. Here is an outline of the steps involved:

Discretize the system of PDEs using a numerical method such as finite difference or finite element method. This will transform the PDEs into a system of algebraic equations.
Combine the discretized PDEs with the algebraic equations to form a system of nonlinear algebraic equations.
Use a numerical solver such as the Newton-Raphson method or a quasi-Newton method to solve the system of nonlinear algebraic equations.
Repeat the process for each time step to obtain a time-dependent solution.

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How to numerically solve a system of coupled partial differential and algebraic equations?

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