1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schemes of nonstationary gas dynamics 529

where the "viscous" pressure w = w(p, v~, h) depends on p, v~ and the step

h. The reader can encounter two types of viscosity:

a) a linear viscosity

(20) w = _ //o h P (av - I ~u 1) ,

2 as as

b) a quadratic viscosity or Neuman's viscosity

(21) w = //o h2 p av (av - lav 1)

2 OS OS OS

where u 0 is the coefficient of viscosity. It follows from the foregoing that

the function w = 0 for av/as > 0 and w i- 0 for av/as < 0, that is, only

within the zone of the shock wave.

Thus, the psevdoviscosity may emerge only within the zone of the

shock wave. The accepted view is to use

or

// av

w=-- -

7] as

w = // (av)2,

7] 0.5

assuming that the coefficient of viscosity depends on the sign of the partial

derivative av/as, so that/) - 0 for av/as> 0.

3. Conservative homogeneous sche1nes. The presence of psevdoviscosity
   1nakes it possible to design homogeneous difference schemes or "through ex-
   ecution" ones, permitting us to reveal the gas distribution caused by shock
   waves. Since the equations of gas dynan1ics express the conservation laws
   of impulse, mass and energy, the scientists wish to have at their disposal
   conservative difference schemes for which the difference equations involved
   reproduce the analogs of these conservation laws on the grid.
   Equations of gas dynamics in integral form are aimed at designing con-
   servative difference schemes by means of the integro-interpolation method:

(22) f (vds-pdt)=O,

( 23) f (i7 ds + v dt) = 0,

(24) f ((E+0.5v^2 )ds-pv dt) = O,