2 Finite Volume Methods

It has already been observed thay many differential equations which we owuld like to solve come from conservation laws which are integrals over volumes. This idea has been carried onto the discretisation of such equations by instead of interpreting $U_j$ as an approximation to a point value, $u(x_j)$, rather

$$U_j\sim{1\over h}\int_{(j-1/2)h}^{(j+1/2)h}u(x){\rm d}x.$$ (49)

In one sense we now reverse the process by which we arrived at the differential equation from a conservation law: take the differential equation and integrate

$$\int_{(j-1/2)h}^{(j+1/2)h}(-u''+\kappa^2u-f){\rm d}x=0.$$ (50)

If we integrate the first term and define

$$\bar f_j={1\over h}\int_{(j-1/2)h}^{(j+1/2)h}f(x){\rm d}x,$$ (51)

then

$$-u'\vert_{(j-1/2)h}^{(j+1/2)h}+\kappa^2hU_j-h\bar f_j=0.$$ (52)

Alternately we can think of applying a conservation argument to the `volume' $((j-{1\over 2})h,(j+{1\over 2})h)$. The difficulty now is to represent the `fluxes' across the faces of the volume at $(j-{1\over 2})h$, $(j+{1\over 2})h$ in terms of the integral quantities, $U_j$. In this simple case we approximate

$$u'((j+{1\over 2})h)\sim {{U_{j+1}-U_j}\over h},$$ (53)

so that we end up with a coefficient matrix which is identical to the finite difference one we derived above, but the right hand side is now a vector of integrals of $f$,

$$({1\over h^2}{\rm\bf K}+\kappa^2{\rm\bf I}^*){\rm\bf U}=\bar{\rm\bf f}.$$ (54)

While this appears very similar to an ordinary finite difference method, if using unstructured meshes in two and three space dimensions (so that the `volumes' are either arbitrary triangles or quadrilaterals) the finite volume method is much easier to apply than conventional finite differences.