Trying to write an ODE solver using Backward Euler with Newton-Raphson method

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Hi, I'm trying to write a function to solve ODEs using the backward euler method, but after the first y value all of the next ones are the same, so I assume something is wrong with the loop where I use NewtonRoot, a root finding function I wrote previously. Do I need to include a separate loop over the Newton-Raphson method? If so, I'm not sure what index I would use.
function [t,y]=BackwardEuler(F,a,b,y0,N,err,imax)
h=(b-a)/N;
y(1)=y0;
t=a:h:b;
syms f(u)
f(u)=u-y0-h*F(u);
df=diff(f,u);
for ii=1:N
y(ii+1)=NewtonRoot(f,df,y(ii),err,imax);
end
end

Risposta accettata

Abraham Boayue
Abraham Boayue il 15 Apr 2018
Alternatively, you can write a function.
function [t,yb,h] = backwardEuler(a,b,N,M,y0,f,fx)
h = (b-a)/N;
t = zeros(1,N);
yb = zeros(1,N);
t(1) = a;
yb(1) = y0;
for j=1:N
x = yb(j);
t(j+1)=t(j)+h;
for i=1:M
x = x-(x-yb(j)-h*f(t(j+1),x))/(1-h*fx(t(j+1),x));
end
yb(j+1)= x;
end
end
f = @(t,x) -0.800*x.^(3/2)+10.*2000 *(1 - exp(-3*t));
fx = @(t,x) -0.800*1.5*x.^(1/2);
a = 0;
b = 2;
N = 250;
M = 20;
err = 0.001;
y0 = 2000;
[t,yb,h] = backwardEuler(a,b,N,M,y0,f,fx);
figure()
plot(t,yb,'linewidth',1.5,'color','b')
grid;
a = title('Backward Euler Method');
set(a,'fontsize',14);
a = ylabel('n');
set(a,'Fontsize',14);
a = xlabel('t(s)');
set(a,'Fontsize',14);
axis([0 0.5 0 2000]),
  1 Commento
Abigail Grein
Abigail Grein il 15 Apr 2018
Do you think there's any way I can use the function I already wrote for Newton's method inside the backward Euler function?

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Più risposte (2)

Abraham Boayue
Abraham Boayue il 13 Apr 2018
I would recommend you writing another function that does the differentiation separately. It would even be best if the function to be differentiated can be done by hand. Can you post the NewtonRoot function? It may give a clue of what's wrong.
  3 Commenti
Abigail Grein
Abigail Grein il 13 Apr 2018
I did think something might have come from trying to make MATLAB do the differentiation I'll try changing that.
Abigail Grein
Abigail Grein il 13 Apr 2018
Hmm differentiating by hand didn't change anything. I think it must be the way I'm using the for loop over the NewtonRoot function.

Accedi per commentare.


Abraham Boayue
Abraham Boayue il 15 Apr 2018
Modificato: Abraham Boayue il 15 Apr 2018
Here are two methods that you can use to code Euler backward formula.
%%Method 1
clear variables
close all
a = 0; b = 0.5;
h = 0.002;
N = (b - a)/h;
M = 20;
n = zeros(1,N);
t = n;
n(1) = 2000;
t(1) = a;
for i=1:N
t(i + 1) = t(i) + h;
x = n(i);
% Newton's implicit method starts.
for j = 1:M
num = x + 0.800*x.^(3/2) *h - 10.*n (1) * (1 - exp(-3*t(i + 1))) *h - n(i) ;
denom = 1 + 0.800*1.5*x.^(1/2) *h;
xnew = x - num/ denom;
if abs((xnew - x)/x) < 0.0001
break
else
x = xnew;
end
end
%Newton's method ends.
n(i + 1) = xnew;
end
figure(1)
plot(t,n,'linewidth',1.5,'color','b')
grid;
a = title('Backward Euler Method');
set(a,'fontsize',14);
a = ylabel('n');
set(a,'Fontsize',14);
a = xlabel('t(s)');
set(a,'Fontsize',14);
axis([0 0.5 0 2000]),
%%Method 2
f = @(t,x) -0.800*x.^(3/2)+10.*2000 *(1 - exp(-3*t));
fx = @(t,x) -0.800*3/2*x.^(1/2);
a = 0;
b = 2;
n = 250;
h = (b-a)/n;
y0 = 2000;
t = zeros(1,n);
yb =zeros(1,n);
t(1)=0;
yb(1)= y0;
% backward Euler method.
for j=1:n
z = yb(j);
t(j+1)=t(j)+h;
for i = 1:20
z = z-(z-yb(j)-h*f(t(j+1),z))/(1-h*fx(t(j+1),z));
end
yb(j+1)=z;
end
figure(2)
plot(t,yb,'linewidth',1.5,'color','r')
grid;
a = title('Backward Euler Method');
set(a,'fontsize',14);
a = ylabel('n');
set(a,'Fontsize',14);
a = xlabel('t(s)');
set(a,'Fontsize',14);
axis([0 0.5 0 2000]),

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