1549301742-The_Theory_of_Difference_Schemes__Samarskii

532 Difference Methods for Solving Nonlinear Equations

The emerging disbalances are characterized by a typical solution with a

wide range of values. For example, they are sufficiently small on sn1ooth

functions, but could grow on solutions varying fastly in time and space

prior to the total energy of the system concerned.

4. Fully conservative sche111es. Other ideas are connected with su<.:cessive
   use of conservation laws and more detailed balances of the kinetic and
   internal energy.
   All the schemes with these properties are called fully conservative
   sche1nes. As a matter of fact, the requirement of the full conservatisn1
   is equivalent to being approximated of both equations (14) and (15) in
   addition to the usual requirements of approximation:
   OE ov OE 07]
   ot = -p os' ot = -p at ·
   Before going further, it will be sensible to introduce more compact
   notations

Pi = Pi+1/2 , 7li = 77i+l/2 , Ei = Ei+l/2 , P =Pi , etc,

1

h (Pi+1/2 - Pi-1/'2) = J3:s,

vVhere there is no danger of confusion, we will omit the sy1nbol "bar" over

p, r;, E. Within these notations, equations (25)-(26) can be reduced to

(28) V t -- -p(" s 1) )

Instead of ( 27), let us consider the scheme generating an approxima-

tion to equation ( 14) capable of describing the law of the internal energy

(29)

In this connection. there arises a four-parameter family of schemes, from
which a fully conservative scheme needs to be selected through the approx-
imations to equations (15) and (10) by appeal to scheme (28)-(29).
vVe will use the obvious relationship

( 30) f(fl) = f(c\') + T (iJ - Ct) ft'

where a; and f3 are arbitrary numbers and f(a)
quantity of interest Et can be discovered from

(31)

a:]+ (1 - ct)f. The