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I'm working on a python code to solve a PDE (Partial Differential Equation) backward in time (i.e. from t = T to t = 0.0). The code is made up of two functions: the first function (def Solve_for_U_values_at_Time_T()) solves a PDE (PDE1) forward in time, and returns an array of values U at a terminal time T (t = 0.0, ..., T). I then want to use this value returned in the first function, in the next function (def Evolve_in_One_Timestep(U)) to solve for a different PDE (PDE2) starting with that value at the terminal time T, going back until I find the value of U at initial time t = 0.0. `Note for k in range (M-2, 0,-1): and for n in range(N-1,-1, -1):`

. See the code below:

```
def Solve_for_U_values_at_Time_T():
# M is the number of grid points, N, number of iterations, Dx spatial step and Dt is the time step
U = zeros(M)
Block of statements
return U
def Evolve_in_One_Timestep(U):
#create an array to store the new values of the solution
new_U = zeros((M,), float32)
#loop over each gridpoint, except for k= M-1 and k = 0
for k in range (M-2, 0,-1):
derivative = -0.25*(U[k+1]**2 - U[k-1]**2)*(Dt/Dx) + 0.5*sqrt(a)*(U[k+1] - 2*U[k] + U[k-1])*(Dt/Dx)
new_U[k] = U[k] + derivative
#applying boundary conditions
new_U[0] = -0.5
new_U[M-1] = 0
return new_U
#initializing the data array
U_Values = Solve_for_U_values_at_Time_T()
for n in range(N-1,-1, -1):
U_Values = Evolve_in_One_Timestep(U_Values)
t = n*Dt
```

`Is my idea correct? Any help/advice will be appreciated.`