How to implement Crank-Nicolson method: update f^n, f^(n+1)

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1

I tried to implement the Crank-Nicolson method for a transport equation.

The Crank-Nicolson method

[tex]M\frac{C^{k+1}-C^k}{\Delta t}+\frac{1}{2}A[ C^{k+1}+C^k]=\frac{1}{2}(f^{k+1}+f^k)[/tex]
or in an incremental form:
[tex]M^(C^{k+1}-C^k)=-\Delta tAC^k+0.5\Delta t(f^{k+1}+f^k)[/tex]
where [tex]M^=M+0.5\Delta t A[/tex]

I don’t know how to represent (f^{k+1}+f^k) in the time loop. I searched on the tutorial, but I didn’t find an example. Please see the attached file.

[code]while t_intermediate < Tend-0.5*dt:
#Update time parameter
t.Set(time+t_intermediate+dt)
f.Assemble()

#RHS: 1/2*dt*(f^(n+1)+f^(n))-dt*A*C^(k)??
res.data = 0.5*dt * (f.vec+f.vec) - dt * a.mat * gfu.vec #wrong
gfu.vec.data += invmstar * res
    
t_intermediate += dt
#print("\r", time + t_intermediate,   end="")
Redraw(blocking=True)

time += t_intermediate[/code]

Could you please tell me how to handle this in NGSolve? Thank you so much.

Attachment: Crank_Nicolson.ipynb

2

Hi dong,

if you need the right hand side from the last time-step, you can simply store this in a vector

f_last = gfu.vec.CreateVector()

and update this at the end of every time-step:

f.Assemble()
f_last.data = f.vec 
while t_intermediate < Tend-0.5*dt:
    t.Set(time+t_intermediate+dt)
    f.Assemble()

    res.data = 0.5*dt * (f.vec+f_last) - dt * a.mat * gfu.vec
    gfu.vec.data += invmstar * res
        
    f_last.data = f.vec
    t_intermediate += dt

Best wishes,
Henry

3

Thank you so much, Henry. I got it.