Discretisation forward difference

At any point with index i, that is at X = iH, the diffusion (1.1) is discretised on the left-hand side in the Euler manner (4.4, or in other words the forward difference formula 3.1) and on the right-hand side with the central three-point approximation (3.41), giving for the iteration going from time T to the next time T + 8T,... 

The writing of O (Ax2) indicates that the discretisation error is proportional to Ax2 and therefore by reducing the mesh size the error approaches zero with the square of the mesh size. The first derivative with respect to time is replaced by the relatively inaccurate forward difference quotient... 

We assume, for a start, the simple diffusion equation 5.12. We have seen in Sect. 5.1, that the normal explicit method, with its forward-difference discretisation of 8c/3t performs rather poorly, with an error of 0(6t). The discrete expression for the second derivative (right-hand side of Eq. 5.12) is better, with its error of 0(h ). Let us now imagine a time t+ig6t at this time, the discretisation... 

For the discretisation expressions, it is again convenient to use the notation in Eq. (13.17) on page 347. At index / = 0, asymmetric one-sided formulae are used, that is, three-point forward differences. 

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