Grid Functions and Difference Operators in Two Dimensions: Cartesian Coordinates

The extension of the definitions in §5.1.3 to two spatial coordinates is, in the Cartesian case, immediate. A grid function $u_{l,m}^{n}$, for $l,m\in{\mathbb{Z}}$, and $n\geq 0$, represents an approximation to a continuous function $u(x,y,t)$, at coordinates $x=lh_{x}$, $y=lh_{y}$, $t=nk$.

The temporal operators behave exactly as those defined in 1D, in §5.1.2, and it is not worth repeating these definitions here. Spatial shift operators, in the $x$ and $y$ directions may be defined as

$$e_{x+}u_{l,m}^{n} = u_{l+1,m}^{n}\qquad e_{x-}u_{l,m}^{n} = u_{l-... ...qquad e_{y+}u_{l,m}^{n} = u_{l,m+1}^{n}\qquad e_{y-}u_{l,m}^{n} = u_{l,m-1}^{n}$$

and forward, backward and centered difference operators as

$$\delta_{x+} \triangleq \frac{1}{h_{x}}\left(e_{x+}-1\right)\approx \frac{\partial}{\part... ...q \frac{1}{2h_{x}}\left(e_{x+}-e_{x-}\right)\approx \frac{\partial}{\partial x}$$ (5.14)

$$\delta_{y+} \triangleq \frac{1}{h_{y}}\left(e_{y+}-1\right)\approx \frac{\partial}{\part... ...q \frac{1}{2h_{y}}\left(e_{y+}-e_{y-}\right)\approx \frac{\partial}{\partial y}$$ (5.15)

Centered second derivative approximations follow immediately as

$$\delta_{xx} = \delta_{x+}\delta_{x-}\approx\frac{\partial^2}{\partial x^2}$$ (5.16)

$$\delta_{yy} = \delta_{y+}\delta_{y-}\approx\frac{\partial^2}{\partial y^2}$$ (5.17)

Quite important in musical sound synthesis applications is the approximation to the Laplacian operator $\Delta$, defined as

$$\Delta = \frac{\partial ^2 }{\partial x^2}+\frac{\partial ^2 }{\partial y^2}$$ (5.18)

There are many ways of approximating this operator. The simplest, by far, is to write

$$\delta_{\Delta}=\delta_{xx}+\delta_{yy} = \Delta +O(h_{x}^2,h_{y}^2)$$ (5.19)

Also important, in the case of the vibrating stiff plate, is the biharmonic operator, or bi-Laplacian, which consists of the composition of the Laplacian with itself, or $\Delta\Delta$. A simple approximation may be given as

$$\delta_{\Delta}\delta_{\Delta}= \Delta\Delta + O(h_{x}^2,h_{y}^2)$$ (5.20)