1-D Parabolic Equation PDE schemes

1-D Heat conduction equation (with no internal heat generation)

$\frac{\delta T}{\delta t} = \alpha \left ( \frac{\delta ^2 T}{\delta x^2} \right )$

Before we dwell any further in the discretization of the above PDE we need to know 3 important terms.

1. Consistency
2. Stability 
3. Convergence

Here a finite difference scheme is said to be consistent when as $\Delta x,\Delta t \rightarrow 0$ , then the scheme  $\rightarrow$ to the original PDE.

A scheme is said to be stable when the ratio of the error propagation from one time step to another is less than $1$.

A scheme is said to converge when the above to conditions are met as well as the solution is close to the actual solution (analytic or numerical) of the PDE .

  

Note : Implicit schemes are mostly unconditionally stable ,while the explicit schemes are conditionally stable or unconditionally unstable. 

Explicit schemes

FTCS scheme

$\frac{T_{i} ^{n+1} -T_{i} ^{n}}{\Delta t} = \frac{T_{i+1} ^{n} -2T_{i} ^{n}+T_{i-1} ^{n}}{(\Delta x)^2}  + O(\Delta t,(\Delta x)^2)$

This scheme is conditionally stable , consistent ,convergent ,second order accurate in space,1st order accurate in time.
For more information on the stability criterion please check the tutorial 1 under the CFD tab.

CTCS scheme

$\frac{T_{i} ^{n+1} -T_{i} ^{n-1}}{2 \Delta t} = \frac{T_{i+1} ^{n} -2T_{i} ^{n}+T_{i-1} ^{n}}{(\Delta x)^2}  + O((\Delta t)^2,(\Delta x)^2)$

This scheme is unconditionally unstable. Hence this scheme is absolutely useless .Therefore the Dufort -Frankel scheme emerged.

Dufort Frankel scheme

$\frac{T_{i} ^{n+1} -T_{i} ^{n-1}}{2 \Delta t} = \frac{T_{i+1} ^{n} -2\left (\frac{T_{i} ^{n+1} +T_{i} ^{n-1}}{2}\right )+T_{i-1} ^{n}}{(\Delta x)^2}  + O((\Delta t)^2,(\Delta x)^2,(\frac{\Delta t}{\Delta x})^2)$

This scheme is unconditionally stable. Hence this scheme is brilliant . The error term can be easily derived , do this as an exercise.

Implicit schemes
BTCS scheme (laasonen scheme)

$\frac{T_{i} ^{n+1} -T_{i} ^{n}}{\Delta t} = \frac{T_{i+1} ^{n+1} -2T_{i} ^{n+1}+T_{i-1} ^{n+1}}{(\Delta x)^2}  + O(\Delta t,(\Delta x)^2)$

Unconditionally stable scheme . This scheme forms a tridiagonal matrix , which can be solved using matrix inversion ,or the Gauss elimination method or the TDMA (thomas) algorithm. For more information on the TDMA algorithm check tutorial 2 under the CFD tab .

Crank Nicolson scheme 

$\frac{T_{i} ^{n+1} -T_{i} ^{n}}{\Delta t} =  \frac{1}{2}\left (\frac{T_{i+1} ^{n+1} -2T_{i} ^{n+1}+T_{i-1} ^{n+1}}{(\Delta x)^2} +\frac{T_{i+1} ^{n} -2T_{i} ^{n}+T_{i-1} ^{n}}{(\Delta x)^2}    \right ) + O((\Delta t )^2,(\Delta x)^2)$

This is a really interesting scheme centered around the time step $n + 1/2$ . As can be seen the above scheme is second order accurate in time as well.
Unconditionally stable scheme . This scheme forms a tridiagonal matrix , which can be solved using matrix inversion ,or the Gauss elimination method or the TDMA (thomas) algorithm. 

Beta Formulation

Finally comes the $\beta $ formulation , this combines all the above mentioned schemes in a single shot .

$\frac{T_{i} ^{n+1} -T_{i} ^{n}}{\Delta t} =  \beta \left (\frac{T_{i+1} ^{n+1} -2T_{i} ^{n+1}+T_{i-1} ^{n+1}}{(\Delta x)^2}  \right )+(1-\beta) \left ( \frac{T_{i+1} ^{n} -2T_{i} ^{n}+T_{i-1} ^{n}}{(\Delta x)^2}   \right )  $

This scheme behaves like an FTCS scheme for $\beta < 1/2$
This scheme behaves like an BTCS scheme for $\beta > 1/2$
This scheme behaves like an Crank Nicolson scheme for $\beta = 1/2$