Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question.

I have a set of three equations in variables $(u,v,w)$ which are one dimensional functions of $x$, and I cannot see how to eliminate the ghost points for the equation for the $w$ variable.

Background

The problem is something like this,

$ \frac{\partial u}{\partial t} = a_u\frac{\partial^2 u}{\partial x^2} + b_u\frac{\partial u}{\partial x} + f_u(x,u,v,w) \\ \frac{\partial v}{\partial t} = a_v\frac{\partial^2 v}{\partial x^2} + b_v\frac{\partial v}{\partial x} + f_v(x,u,v,w) \\ \frac{\partial w}{\partial t} = a_u\frac{\partial u}{\partial x} +a_v\frac{\partial v}{\partial x} + f_w(x,u,v,w) \\ $

The set of equations describe a two species advection-diffusion problem where the third equation couples to the other two. I am having problems applying Neumann boundary conditions to the third equation. Notice that the third equation does not contain a differential of $w$, but it does contain differentials of both $u$ and $v$.

In discretizated form (only showing the $w$ equation, where $\Delta t$ is the time step and $s_u$ and $s_v$ are the coefficients of the equations after discretization),

$ w_j^{n+1} - s_u\beta(u_{j+1}^{n+1} - u_{j-1}^{n+1}) - s_v\beta(v_{j+1}^{n+1} - v_{j-1}^{n+1}) = s_u(1-\beta)(u_{j+1}^{n} - u_{j-1}^{n}) + s_v(1-\beta)(v_{j+1}^{n} - v_{j-1}^{n}) + w_j^n + \Delta tf_{j}^n $

The problem

The above is a Crank-Nicolson discretization, but for the boundary points let's use an implicit scheme (setting $\beta=1$),

$ w_1^{n+1} - s_u(u_{2}^{n+1} - u_{0}^{n+1}) - s_v(v_{2}^{n+1} - v_{0}^{n+1}) = w_1^n + \Delta tf_{1}^n $

The ghost points, $u_0^{n+1}$ and $v_0^{n+1}$ can be eliminated, by using the boundary conditions for the $u$ and $v$ variables. However, applying Neumann boundary conditions to the $w$ variable gives,

$ \frac{\partial w}{\partial x} = \sigma |_{j=1} \\ \frac{1}{2 \Delta x} (w_{2}^{n+1} - w_{0}^{n+1} ) = \sigma $

The problem is that the boundary equation cannot be substituted into the equation for $j=1$ because it does not contain at term for $w_{0}^{n+1}$.

Question

Do you have any suggestions on how I can apply Neumann boundary conditions to this equation?

Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question.

I have a set of three equations in variables $(u,v,w)$ which are one dimensional functions of $x$, and I cannot see how to eliminate the ghost points for the equation for the $w$ variable.

Background

The problem is something like this,

$ \frac{\partial u}{\partial t} = a_u\frac{\partial^2 u}{\partial x^2} + b_u\frac{\partial u}{\partial x} + f_u(x,u,v,w) \\ \frac{\partial v}{\partial t} = a_v\frac{\partial^2 v}{\partial x^2} + b_v\frac{\partial v}{\partial x} + f_v(x,u,v,w) \\ \frac{\partial w}{\partial t} = a_u\frac{\partial u}{\partial x} +a_v\frac{\partial v}{\partial x} + f_w(x,u,v,w) \\ $

The set of equations describe a two species advection-diffusion problem where the third equation couples to the other two. I am having problems applying Neumann boundary conditions to the third equation. Notice that the third equation does not contain a differential of $w$, but it does contain differentials of both $u$ and $v$.

In discretizated form (only showing the $w$ equation, where $\Delta t$ is the time step and $s_u$ and $s_v$ are the coefficients of the equations after discretization),

$ w_j^{n+1} - s_u\beta(u_{j+1}^{n+1} - u_{j-1}^{n+1}) - s_v\beta(v_{j+1}^{n+1} - v_{j-1}^{n+1}) = s_u(1-\beta)(u_{j+1}^{n} - u_{j-1}^{n}) + s_v(1-\beta)(v_{j+1}^{n} - v_{j-1}^{n}) + w_j^n + \Delta tf_{j}^n $

The problem

The above is a Crank-Nicolson discretization, but for the boundary points let's use an implicit scheme (setting $\beta=1$),

$ w_1^{n+1} - s_u(u_{2}^{n+1} - u_{0}^{n+1}) - s_v(v_{2}^{n+1} - v_{0}^{n+1}) = w_1^n + \Delta tf_{1}^n $

The ghost points, $u_0^{n+1}$ and $v_0^{n+1}$ can be eliminated, by using the boundary conditions for the $u$ and $v$ variables. However, applying Neumann boundary conditions to the $w$ variable gives,

$ \frac{\partial w}{\partial x} = \sigma |_{j=1} \\ \frac{1}{2 \Delta x} (w_{2}^{n+1} - w_{0}^{n+1} ) = \sigma $

The problem is that the boundary equation cannot be substituted into the equation for $j=1$ because it does not contain at term for $w_{0}^{n+1}$.

Question

Do you have any suggestions on how I can apply Neumann boundary conditions to this equation?