Derivative operators

Expressions for the derivative operators, such as gradient, divergence, curl, Laplacian, etc., are obtained by applying the Divergence Theorem to a differential volume increment bounded by coordinate surfaces. The gradient operator then leads to the expression of contravariant base vectors in terms of the covariant base vectors, and to the contravariant metric tensor as the inverse of the covariant metric tensor.

By Divergence theorem,

$$\int\int\int_{V}\overline{\nabla}.\overline{A}.dV = \int\int_{\delta V}\overline{A}.\overline{dS}$$ (3.10)

for any tensor $\overline{A}$ , where $\overline{dS}$ is the outward-directed normal area element to the closed surface $\delta V$ enclosing the volume V. For a differential surface element lying on a coordinate surface we have,

$$\overline{dS}^{i}=\pm \left(\overline{a}_{j} \times \overline{a}_{k}\right) d\xi^{j}d\xi^{k}$$ (3.11)

with the choice of sign being dependent on the location of the volume relative to the surface. Then considering a differential element of volume, V, bounded by six faces lying on coordinate surfaces, as shown in the figure, we have from 3.8 and 3.9,

$$\int\int\int_{V}\overline{\nabla}.\overline{A}\sqrt{g}d\xi^i... ...{a}_{j} \times \overline{a}_{k}\right) d\xi^{j}d\xi^{k}\right)$$ (3.12)

where the notation $\delta S^{i}_{+}$ and $\delta S^{i}_{-}$ indicates the element on $\xi^{i}$ two sides of the which $\xi^{i}$ is constant and which are located at larger and smaller values, respectively, of $\xi^{i}$. Here, as usual, the indices (i,j,k) are cyclic.

Proceeding to the limit as the element of volume shrinks to zero we then have an expression for the divergence:

$$\overline{\nabla}.\overline{A}={1 \over \sqrt{g}}\sum_{i=1}^... ...\times \overline{a}_{k} \right). \overline{A}\right)_{\xi^{i}}$$ (3.13)

This expression can be expanded by parts and noting that

$$\sum_{i=1}^{3}\left(\overline{a}_{j} \times \overline{a}_{k} \right)_\xi^{i}=0$$ (3.14)

the expression for divergence can be also written as

$$\overline{\nabla}.\overline{A}={1 \over \sqrt{g}}\sum_{i=1}^... ...a}_{j} \times \overline{a}_{k} \right). \overline{A}_{\xi^{i}}$$ (3.15)

Although the equations 3.13 and 3.15 are equivalent expressions for the divergence, because of the identity 3.14, the numerical representations of these two forms may not be equivalent. The form given by Eq. 3.13 is called the conservative form, and that of Eq. 3.15, where the product derivative has been expanded and Eq. 3.14 has been used, is called the non-conservative form. Recalling that the quantity $\left(\overline{a}_j \times \overline{a}_k\right)$ represents an increment of surface area (cf.Eq. 3.7), so that $\left(\overline{a}_j \times \overline{a}_k\right).\overline{A}_{\xi^{i}}$ is a flux through this area, it is clear that the difference between the two forms is that the area used in numerical representation of the flux in the conservative form, Eq. 3.13, is the area of the individual sides of the volume element, but in the nonconservative form, a common area evaluated at the center of the volume element is used. The conservative form thus gives the telescopic collapse of the flux terms when the difference equations are summed over the field, so that this summation then involves only the boundary fluxes. This would seem to favor the conservative form as the better numerical representation of the net flux through the volume element.

It is important to note that since the conservative form of the divergence, and of the gradient and Laplacian to follow, is obtained directly from the closed surface integral in the Divergence Theorem, the use of the conservative difference forms for these derivative operators is equivalent to using difference forms for that closed surface integral. Therefore the finite volume difference formulation can be implemented by using these conservative forms directly in the differential equations of motion without the necessity of returning to the integral form of the equations of motion.

Eq. 3.10 is also valid with $\overline{A}$ replaced by a scalar, and the dot product replaced by simple operation on the left and multiplication on the right. Therefore the conservative and non-conservative expressions for the gradient also follow directly from Eq. 3.13 and 3.15 as

$$\overline{\nabla}A={1 \over \sqrt{g}}\sum_{i=1}^{3}\left( \l... ...line{a}_{j} \times \overline{a}_{k} \right) A\right)_{\xi^{i}}$$ (3.16)

and

$$\overline{\nabla}A={1 \over \sqrt{g}}\sum_{i=1}^{3} \left(\overline{a}_{j} \times \overline{a}_{k} \right) A_{\xi^{i}}$$ (3.17)

The expressions for the Laplacian then follow from Eq. 3.13 or 3.15, with $\overline{A}$ replaced by $\overline{\nabla}A$ from Eq. 3.16 or 3.17. Thus the conservative form is

$$\overline{\nabla}.\overline{\nabla}A= {1 \over \sqrt{g}}\sum... ...es \overline{a}_{n}\right)A\right\}_{\xi^{l}}\right]_{\xi^{i}}$$ (3.18)

and the non-conservative form is

$$\overline{\nabla}.\overline{\nabla}A= {1 \over \sqrt{g}}\sum... ...m} \times \overline{a}_{n}\right) A_{\xi^{l}}\right]_{\xi^{i}}$$ (3.19)