The Operator ${\mathcal L}[\cdot,\cdot]$

The operator ${\mathcal L}[\cdot,\cdot]$ is defined, in Cartesian coordinates, as

$${\mathcal L}[\alpha,\beta] = \frac{\partial^2\alpha}{\partial x^2... ...tial^2\alpha}{\partial x\partial y}\frac{\partial^2\beta}{\partial x\partial y}$$

Clearly,

$${\mathcal L}[\alpha,\beta] = {\mathcal L}[\beta,\alpha]$$ (11.2)

The operator ${\mathcal L}[\cdot,\cdot]$ is bilinear, or linear in either one of its arguments if the other is held fixed. Considering the first argument, for any constants $c_{1}$ and $c_{2}$,

$${\mathcal L}[c_{1}\alpha_{1}+c_{2}\alpha_{2},\beta] = c_{1}{\mathcal L}[\alpha_{1},\beta]+c_{2}{\mathcal L}[\alpha_{2},\beta]$$ (11.3)

for any functions $\alpha_{1}$, $\alpha_{2}$ and $\beta$. (It will be linear in the second argument as well by (11.2).) The bilinearity property also implies that

$$\dot{\left({\mathcal L}[\alpha,\alpha]\right)} = 2{\mathcal L}[\dot{\alpha},\alpha]$$ (11.4)

Another important property (for lack of a better term, it will be called ``triple self-adjointness" in this article) is that for any three functions $\alpha$, $\beta$ and $\gamma$ defined over the quarter plane ${\mathcal A}$,

$$\iint_{{\mathcal A}}\alpha {\mathcal L}[\beta,\gamma]d\sigma = \iint_{{\mathcal A}}{\mathcal L}[\alpha,\beta]\gamma d\sigma$$ (11.5)

$$+ \int_{0}^{\infty}\alpha_{y}\beta_{xx}\gamma-\alpha\left(\beta_{... ...{y}-\alpha_{x}\beta_{xy}\gamma +\alpha\left(\beta_{xy}\gamma\right)_{x}dx\notag$$ (11.6)

$$+ \int_{0}^{\infty}\alpha_{x}\beta_{yy}\gamma-\alpha\left(\beta_{... ...{x}-\alpha_{y}\beta_{xy}\gamma +\alpha\left(\beta_{xy}\gamma\right)_{y}dy\notag$$ (11.7)

In designing a conservative numerical scheme for (11.1), it is crucial that discrete analogues of the properties (11.4) and (11.5) be maintained. The bilinear property (11.3) is the key to efficient computer realizations, as will be discussed in Section 11.2.2.