31.1.1 Lattice and continuum operators

A good reference for the methods used in this subsection is chapter 2 from the book by Mitchell and Griffiths (1980).

Consider the continuum field $\psi$ evaluated at an arbitrary lattice point

$$\psi^{n}_{i} = \psi(x_{i},t_{n}).$$ (31.2)

Using a Taylor series expansion, the relation between $\psi^{n}_{i}$and $\psi^{n}_{i + p}$, where $x_{p} = p \, \Delta x$, is given by

$$\psi^{n}_{i + p} \left( 1 + p \, \Delta x \, \partial_{x} + \frac{1}{2!} \, (p \, ... ...{1}{3!} \, (p \, \Delta x)^{3} \partial^{(3)}_{x} + \ldots \right) \psi^{n}_{i}$$ =

$$\exp( p \, \Delta x \, \partial_{x}) \, \psi^{n}_{i}.$$ = (31.3)

The linear operator $\exp( p \, \Delta x \, \partial_{x})$ can be thought of as a spatial translation operator; a terminology familiar to those having studied quantum mechanics. Similarly, the temporal translation operator $\exp(q \, \Delta t \, \partial_{t})$ connects $\psi^{n}_{i}$ to another point in time $t_{q} = q \, \Delta t$through the relation

$$\psi^{n + q}_{i} = \exp(q \, \Delta t \, \partial_{t}) \, \psi^{n}_{i}.$$ (31.4)

In the following, it will prove useful to derive relations between the continuum differential operators $\partial_{x}$ and $\partial_{t}$ and various finite difference or lattice operators. To start, consider the familiar centered difference spatial operator as defined by

$$\delta^{C}_{x} \psi_{i}^{n} = \frac{\psi_{i+1}^{n} -\psi_{i-1}^{n}}{2 \Delta x}.$$ (31.5)

Using the spatial translation operators, this relation takes the form

$$\delta^{C}_{x} \psi_{i}^{n} = \left( \frac{\sinh(\Delta x \, \partial_{x}) }{\Delta x} \right) \psi_{i}^{n}.$$ (31.6)

Since $\psi_{i}^{n}$ is arbitrary, this equation yields the relation between the centered difference lattice operator and the continuous partial derivative

$$\Delta x \, \delta_{x}^{C} = \sinh(\Delta x \, \partial_{x}).$$ (31.7)

Inverting this relation yields

$$\Delta x \, \partial_{x} = \sinh^{-1}(\Delta x \, \delta^{C}_{x} ).$$ (31.8)

Similar relations hold for the temporal centered difference, or leap frog, operator

$$\Delta t \, \delta_{t}^{C} \sinh(\Delta t \, \partial_{t})$$ = (31.9)

$${\Delta t} \, \partial_{t} \sinh^{-1}(\Delta t \, \delta^{C}_{t} ).$$ = (31.10)

The forward difference lattice operator is also quite common

$$\delta^{F}_{x} \psi_{i}^{n} \frac{\psi_{i+1}^{n} -\psi_{i}^{n}}{\Delta x}$$ =

$$[ \exp(\Delta x \, \partial_{x}) - 1 ] \, \psi_{i}^{n}.$$ = (31.11)

This relation leads to the operator equalities

$$\Delta x \, \delta^{F}_{x} \exp(\Delta x \, \partial_{x}) - 1$$ = (31.12)

$$\Delta x \, \partial_{x} \ln(1 + \Delta x \, \delta_{x}^{F}),$$ = (31.13)

with analogous results for the temporal forward difference operator. The relation between other finite difference operators and the continuum operator can be derived similarly.