Discretization (BTCS scheme)

[Wee]

How to discretize the following model using backward time forward space scheme?

$$\frac{\partial c} {\partial t} = \frac {\partial^{\smash{2}} c} {\partial x^{\smash{2}}} + \rho c$$
The discretized equation is $$c_{i, j} = -Qc_{i-1, j+1} + Pc_{i, j+1} - Qc_{i+1, j+1}$$where $$P = \frac {1+2\beta} {1+\alpha}$$ and $$Q = \frac {\beta} {1+\alpha}$$ with $$\beta = \frac {\Delta t} {(\Delta x)^ {\smash{2}}}$$ and $$\alpha = \frac {\rho L^{2}} {D} \Delta t$$
I tried to substitute the formula into the model but couldn't get the answer and I don't know that is L. Please help me!!!

 

[Deleted member 4993]

How to discretize the following model using backward time forward space scheme?

$$\frac{\partial c} {\partial t} = \frac {\partial^{\smash{2}} c} {\partial x^{\smash{2}}} + \rho c$$
The discretized equation is $$c_{i, j} = -Qc_{i-1, j+1} + Pc_{i, j+1} - Qc_{i+1, j+1}$$where $$P = \frac {1+2\beta} {1+\alpha}$$ and $$Q = \frac {\beta} {1+\alpha}$$ with $$\beta = \frac {\Delta t} {(\Delta x)^ {\smash{2}}}$$ and $$\alpha = \frac {\rho L^{2}} {D} \Delta t$$
I tried to substitute the formula into the model but couldn't get the answer and I don't know that is L. Please help me!!!

When you substituted the formula into the model,

What did you get?​

​

Please share your work!

 

[Wee]

When you substituted the formula into the model,

What did you get?​

​

Please share your work!

Hi, thank you for replying to me. Here is my work and I don't know where the L comes from.