# Hi, all. Could you help me to check my code?

330

views

-2

Hi, all,

I am implementing Gauge method for Naver-Stokes equations, see

https://web.math.princeton.edu/~weinan/pdf%20files/gauge%20method.pdf in details.

The Gauge method contains three steps, as follows.

$m^i,\quad u^i,\quad\phi^i$

with the boundary conditions as follows

$m^{i+1}=\left(\nabla\phi^i\cdot\tau\right)\tau,\quad\quad on\quad\partial\Omega$

with pure Neumann boundary condition

$\frac{\partial\phi^{i+1}}{\partial n}=0$∂

Could you tell me how to implement the procedure?

I am implementing Gauge method for Naver-Stokes equations, see

https://web.math.princeton.edu/~weinan/pdf%20files/gauge%20method.pdf in details.

The Gauge method contains three steps, as follows.

$m^i,\quad u^i,\quad\phi^i$

`m`^{i},`u`^{i},`ϕ`^{i}are known as the intial values- find $m^{i+1}$
`m`^{i+1}

`m`^{i+1}−`m`^{i}`d``t`−`μ`Δ`m`^{i+1}+(`u`^{i}·∇)`u`^{i}=`ƒ`,`i``n`Ωwith the boundary conditions as follows

$m^{i+1}=\left(\nabla\phi^i\cdot\tau\right)\tau,\quad\quad on\quad\partial\Omega$

`m`^{i+1}=(∇`ϕ`^{i}·`τ`)`τ`,`o``n`∂Ω- find $\phi^{i+1}$
`ϕ`^{i+1}

`ϕ`^{i+1}=∇·`m`^{i+1}, in $\Omega$Ωwith pure Neumann boundary condition

$\frac{\partial\phi^{i+1}}{\partial n}=0$∂

`ϕ`^{i+1}∂`n`=0- find $u^{i+1}$
`u`^{i+1}

`u`^{i+1}=`m`^{i+1}−∇`ϕ`^{i+1}Could you tell me how to implement the procedure?

Please login to add an answer/comment or follow this question.

Could you tell me how to implement the boundary condition of $m^{i+1}$

m^{i+1}for the first step?I did my best, however, it still does not work.

Best,

Hamilton

$m^{i+1}=\left(\nabla\phi^{i+1}\cdot\tau\right)\tau$

m^{i+1}=(∇ϕ^{i+1}·τ)τThe related topic are can be seen,

https://www.allanswered.com/post/mgml/how-to-solve-poisson-equation-with-tangential-boundary-condition/