1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative schemes^151

conservation laws (of heat, mass, momentun1, energy, etc.). Usually the

derivation of a differential equation of mathematical physics is on an in-

tegral relation (a balance equation) which expresses such a conservation

law for a small volume. Letting the volume to zero and assuming that the

derivatives involved in the balance equation are continuous, one can write

down the appropriate differential equation.

From a physical point of view, the finite difference method is mostly

based based on the further replacement of a continuous medium by its

discrete model. Adopting those ideas, it is natural to require that the

principal characteristics of a physical process should be in full force. Such

characteristics are certainly conservation laws. Difference schemes, which

express various conservation laws on grids, are said to be conservative or

divergent. For conservative schemes the relevant conservative laws in the

entire grid domain (integral conservative laws) do follow as an algebraic

corollary to difference equations.

In this view, it see111s reasonable to construct conservative difference

sche111es with the aid of balance equations for an elementary volume (cell) of

a grid domain. Integrals and derivatives involved in these equations should

be replaced by approximate difference expressions, thereby completing the

design of a homogeneous difference scheme. The way of obtaining homoge-

neous difference schemes is called the integro-interpolational method

or balance method.

In what follows the illustration of this method is concerned with equa-

tion ( 1) from (Section 1) capable of describing the stationary distribution

of temperature over a homogeneous bar 0 < x < 1. The equation of the

heat balance can be written on the segment xi_ 112 < x < xi+ 1 ; 2 as

xi+1/2 xi+1/2

( 11) wi-1/2 - wi+1/2 + j f(x) dx = j q(x) u(x) dx,

Xi-1/2 Xi-1/2

w -- -k u' '

where w(x) is the heat flow, q(x) u(x) is the capacity of heat sinks (sources
for q < 0) proportional to the temperature and J(x) is the distribution
density of external heat sources (sinks).
The existence of a heat sink owes a debt to the heat exchange with the
external environment on the lateral surface of the bar. The quantity wi 112
is equal to the amount of heat being supplied to the segment X; 112 < x <
xi+ 1 ; 2 through the cross-section x = xi_ 112 , while wi+ 1 ; 2 refers in a similar
fashion to the heat transfer from this segment through the cut x = X;+ 1 ; 2.
The third member on the left-hand side of ( 11) reflects the amount of