MATLAB: Eulerian Position and Velocity Updates – Compressible Fluid Dynamics

compressible flowcomputational fluid dynamicsfluid dynamics

Hello Everyone,
It is my first time writing in this forum, so please be patient with me 🙂
I am writing a code following Smoothed Particle Computation (SPC) method for solving a Compressible Fluid Dynamics Problem.
In my code I got to a point where I need to do the following:
Basically, say at time = 0, I have a certain velocity u, a position x and acceleration a.
Now, I want to update u and x for an amount of time steps, using the following fomulae:
u (t+1) = u(t) + acceleration*dt
x(t+1) = x(t) + u(t+1)*dt
t = original time/time index
t+1 = new time
dt = 1 (for example), acceleration = same for every time step. In my case I would like to have 10 time steps.
How do I do it?
Thanks, Fed
Here is my code so far, just for fashion :
clc, clear
N=61; sigma = 5; m = 1;  a=1;
j = 1:61; 
x_j = j - 1 + 0.5*sign((N+1)/2 - j).*(1-exp(-0.2.*( j-(N+1)./2).^2) )
    for x_i = [1:61]
    ro = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j).^2)./sigma^2)
    del_ro = (2*m)/(sqrt(pi)*sigma^3).*(x_i-x_j).^2.*exp((-(x_i-x_j).^2)./sigma^2)
    u_j = (-(a^2.*log(ro)))./2
    acc = (- a^2./ro).*del_ro
    end
for k = 1:10
u_j_new(k+1) = u_j(k) +acc(k)
     x_j_new(k+1) = [x_j(k).*u_j_new(k)]
     ro_new(k+1) = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j_new).^2)./sigma^2)
end

Best Answer

Are u_j x_j acc vectors corresponding to 61 different points? Do you want to treat each point individually? In other words, do you want to find the new position,velocity, and ro of each index?
In that case you would do
for k = 1:10  
u_j = u_j +acc
x_j = x_j(k)+u_j_new
ro = m/(sqrt(pi)*sigma).*exp((-(x_i-x_j).^2)./sigma^2)
end
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