# Periodic Neumann Boundary Conditions (2D Channel)
**URL:** https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855
**Category:** General Discussion
**Created:** [December 27, 2023, 1:04pm UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855 "2023-12-27T13:04:23Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@C98](https://community.freefem.org/u/C98)
#### Post date: [December 27, 2023, 1:04pm UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/1 "2023-12-27T13:04:23Z")
Hi, I am new to FreeFem++, and I am trying to simulate the unsteady Navier-Stokes equations in a 2D channel [0,1]\times[0,1] with some periodic boundary conditions. I denote the tangential velocity by u\_1, the normal velocity by u\_2, and the pressure by Pr.
The boundary conditions are
u\_1(0, y) = u\_1 (1, y),
u\_2(0, y) = u\_2 (1, y),
\partial\_xu\_2(0, y) = \partial\_xu\_2(1, y),
Pr(0, y) = Pr(1, y) + a,
u\_1(x, 0) = 0,
u\_1(x, 1) = 0,
u\_2(x, 0) = f(x,t),
u\_2(x, 1) = 0,
where a is some nonnegative constant, and f(x,t) is a given function.
To deal with the periodic Dirichlet boundary conditions, I use the ‘periodic’ function in the definition of the element space. Since we cannot mix periodic and non-periodic elements, I define p := Pr - Pbar, where Pbar = -ax+1, so that p is periodic, and I use p in the variational formulation.
My big problem now is how to implement the periodic Neumann boundary condition? In my code, everything runs without problem but I did not specify anywhere the periodic Neumann boundary condition, so I guess the simulated solution is just wrong. How to specify periodic Neumann boundary conditions?
My code is below (with f=0 and a=0.5):
```auto
// Parameters
real dt = 0.5; // Time step
int meshSize = 50;
real L= 1;
real RE = 1000;
// Mesh boundary
border b1(s = 0, 1){x=s*L; y=0; label=1;}
border b2(s= 0, 1){x=L; y= s; label=2;}
border b3(s= 0, 1){x=L-s*L; y= 1; label=3;}
border b4(s= 0, 1){x=0; y= 1-s; label=4;}
func perio=[[4,y],[2,y]];
mesh Th = buildmesh(b1(meshSize) + b2(meshSize) + b3(meshSize) + b4(meshSize));
plot(Th, wait=true);
fespace Vh(Th, P1, periodic=perio);
Vh u1, v1, u2, v2, p, q, up1, up2, pp;
int i=0;
real alpha=1/dt;
real nu = 1/RE;
real epsi = 0.00001;
real a = 0.5;
problem NS (u1, u2, p, v1, v2, q, solver=Crout, init=i)
= int2d(Th)(
alpha*(u1*v1 + u2*v2)
+ nu * (
dx(u1)*dx(v1) + dy(u1)*dy(v1)
+ dx(u2)*dx(v2) + dy(u2)*dy(v2)
)
- p*q*(epsi)
- p*dx(v1) - p*dy(v2)
- dx(u1)*q - dy(u2)*q
)
+ int2d(Th)(a*x*q*(epsi) + a*x*dx(v1) + a*x*dy(v2))
- int2d(Th)(1*q*(epsi)+1*dx(v1)+1*dy(v2))
+ int2d(Th)(
- alpha*convect([up1,up2],-dt,up1)*v1
- alpha*convect([up1,up2],-dt,up2)*v2)
+ on(1, 3, u1=0)
+ on(3, u2=0)
+ on(1, u2=0);
u1 = -cos(pi*x/4.0)*exp(-y*y/2.0);
u2 = 0;
p = -cos(pi*x/4.0)*exp(-y*y/2.0)+a*x-1;
for (i = 0; i <= 500000; i++){
// Update
pp = p;
up1 = u1;
up2 = u2;
//vort = dx(u2)-dy(u1);
// Solve
NS;
// Plot
plot([u1,u2], fill=true);
}
```
Thank you in advance for your answers,
---
### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [December 27, 2023, 3:57pm UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/2 "2023-12-27T15:57:24Z")
1. just a first remark, the characteristics method in not correctly implemented with periodic probem ( no way to day).
2. no problem with Periodic Neumann become the function u will be peridic so dx(u)
is also periodic.
3. the characteristic method need a CFL close too 1 , a here the velocity is proportional to RE.
4. p is defined through a constant so int2d(Th)(p) == 0 to set this constante.
Some correction but you have to change the method for do the Non linear part.
[perio-NS.edp](https://community.freefem.org/uploads/short-url/uq2cK2b1aki5WmvF1YSy68WfOET.edp) (1.4 KB)
---
### Author:  [@C98](https://community.freefem.org/u/C98)
#### Post date: [December 29, 2023, 12:18pm UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/3 "2023-12-29T12:18:07Z")
Thank you very much for your answer! It is very clear now. I will see how to implement the convective part.
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### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [December 29, 2023, 3:48pm UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/4 "2023-12-29T15:48:21Z")
you can just do for the term u.\nabla u the previous u like up.\nabla u ,
in this cas you mush rebuild the matrix in the loop so init = 1
---
### Author:  [@stergopilot](https://community.freefem.org/u/stergopilot)
#### Post date: [November 24, 2025, 8:52am UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/5 "2025-11-24T08:52:25Z")
Dear Prof. Hecht,
The **characteristics method** is **not correctly implemented** when combined with **periodic boundary conditions** in FreeFem++. Is it fixed in the recent development?
Thank you.
---
### Author:  [@frederichecht](https://community.freefem.org/u/frederichecht)
#### Post date: [January 28, 2026, 10:30am UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/6 "2026-01-28T10:30:05Z")
No, because it is too hard to do.
---
### Author:  [@stergopilot](https://community.freefem.org/u/stergopilot)
#### Post date: [January 29, 2026, 2:06am UTC](https://community.freefem.org/t/periodic-neumann-boundary-conditions-2d-channel/2855/7 "2026-01-29T02:06:21Z")
OK. Thank you so much.