# FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\rvert_{x=1}=0, \quad t \in (0,1)$$ $$u(0,x)=u_0, \quad x \in (0,1)$$ $$u(0,0) =\mu_0, \quad u(0,1)=\mu_1,$$ where $$f,g,u_0$$ are given functions and $$u_1,u_2$$ are constant. I'm struggling in the discretization to obtain the corresponding linear system.

For the first equation taking $$u_j^n=u(t_n,x_j)$$ and using explicit Euler scheme, I found $$u_j^{n+1}=ru_{j-1}^n+(1-2r)u_j^n +ru_{j+1}^n,$$ with $$r=\dfrac{\Delta t}{(\Delta x)^2}$$, but I don't know how to discretize the second and third equations. For the $$x$$-derivatives I think I should use centered formula by adding two ghost points as in Neumann b.c, and the same for time derivatives but this is a bit complicated and yields much of equations.

Maybe I should treat the equations like a coupled system through the boundary but I don't know how to do that. What is the right FDM discretization. Thanks for any help.

PS: My background to FDM is very simple (just theoretical) and this is my first time to simulate PDE. So, what is your advice to this in a right way? (methods, languages, programs...etc).

• Do you have 4 boundary conditions for a second order equation? – nicoguaro Jul 26 '19 at 12:16
• @nicoguaro Yes, you can view this as a system of equations coupled through the boundary $u$ and $u_b$. In fact the first one is in the domain and 2nd+3d are just one equation with the normal derivative on the boundary. – Migalobe Jul 26 '19 at 12:22
• As far as I know you need 2. Making this system overdetermined. – nicoguaro Jul 26 '19 at 12:25
• it's just one b.c (with corresponding initial conditions) see this system – Migalobe Jul 26 '19 at 12:29
• That's not what you wrote in the question. – nicoguaro Jul 26 '19 at 14:18