Hi ayushman,
I understand that you are trying to solve the first-order wave equation using two different numerical time-stepping methods: Euler's method and the Runge-Kutta method.
The Euler method is a explicit time-stepping approach. To update each grid point 'i' at the subsequent time step 'n+1', you can employ the following formula within your code:
% Euler Method
for n = 1:T-1
for i = 2:Nx
U(i, n+1) = U(i, n) - c * (U(i, n) - U(i-1, n));
end
end
The Runge-Kutta method is a more accurate method than Euler's method. The second-order Runge-Kutta method (RK2), also known as the midpoint method, can be applied to the wave equation as illustrated below:
% Runge-Kutta Method (RK2)
for n = 1:T-1
for i = 2:Nx
k1 = -c * (U(i, n) - U(i-1, n));
k2 = -c * (U(i, n) + 0.5 * k1 - U(i-1, n) - 0.5 * k1);
U(i, n+1) = U(i, n) + k2;
end
end
Please follow the provided algorithmic outlines and make necessary adjustments for the boundary conditions as you proceed with your implementation.
I hope this helps!
