Next:Explicit Formulae: Up:Main Previous:Differentiation Formulas:

Derivation of an exact difference formula

Consider the linear parabolic PDE

$$\frac{\partial u}{\partial t}=L(t,x,D,D^{2})u$$ (1)

where the operator L is linear and $$D=\frac{\partial}{\partial x}$$
The difference formulae involving two adjacent time levels are obtained from the Taylor expansion

$$u(x,t+k) =$$

$$=$$

If we now put $x=mh,$    $t=nk$ and     $u(mh,nk)=u^{n}_{m},$ then

Now an exact formula connecting D and $$\delta_{x},$$ the central difference operator in the x-direction, is

$$D=\frac{2}{h}\sinh^{-1}\frac{\delta x}{2}=\frac{1}{h}\left(\delta... ...}.3!}\delta^{3}_{x}+ \frac{1^{2}.3^{2}}{2^{4}. 5!}\delta^{5}_{x}+ ..... \right)$$ (3)

If equation (3) is used to eliminate D in terms of $\delta x$ in equation (2), the exact difference replacement is given by

All difference formulae in common use for solving equation (1) are approximations of equation (4).


Previous:Differentiation Formulas