Explicit Eulers Method for time advancement

Hello I am trying to use explicit Euler for time advancement and the second-order centraldifference scheme for the spatial derivative, solve the equation to steady

state on a uniform grid. Plot the exact and numerical steady solutions for Nx = 10, 20.

𝜕𝑇/𝜕𝑡 = 𝛼*( 𝜕^2𝑇/𝜕𝑥^2) + 𝑆(𝑥) on the boundary of 0 ≤ 𝑥 ≤ 𝐿𝑥 The initial and boundary conditions are 𝑇(𝑥, 0) = 0 𝑇(0,𝑡) = 0 𝑇(𝐿𝑥,𝑡) = 𝑇steady(𝐿𝑥) Take 𝛼 = 1, 𝐿𝑥 = 15, and 𝑆(𝑥) = −(𝑥 2 − 4𝑥 + 2)𝑒 −𝑥 . The exact steady solution is 𝑇steady(𝑥) = 𝑥 2𝑒 −𝑥

heres the code I have can someone explain where I went wrong

alpha =0;
x = 0;
n =10;
T(0)4ess = 0;
h=0.1;
s(x)=-(x^2-4*x+2)*exp^(-x);
for i=1:n
    T(i+1)=T(i) + h;
    T(i)^(n+1)=T(i+1)^n+((alpha*h)/(x+h)^2)*(T(i+1)^n-2T(i)^n+T(i-1)^n)+h*s(x);
    x = x +1;
    h = h +0.1;
end

plot(x,T);
grid on;

[darova]

Can you please write difference scheme in LaTeX?

Answer 1

[Alan Stevens]

T(i)^(n+1)

This will raise T(i) to the (n+1)th power!

You need another loop for time (say j = 1:something), then you can refer to T at position i and time j as

T(i,j)

On the right hand side

((alpha*h)/(x+h)^2)*(T(i+1)^n-2T(i)^n+T(i-1)^n)

should be

((alpha*h)/dx^2)*(T(i+1,j)-2T(i,j)+T(i-1,j))

where dx is the spatial interval (dx = Lx/n).

h is the timestep, so don't update it in the loop!