1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 85

Evidently, a special attention in approximating problem (51) is being

paid to the difference forrn of writing the initial condition for the derivative

au/at. On any equidistant in x and t grid w1i 7 with steps hand T (see

Section 2.1) the simplest approxirnation 1t 1 (x, 0) = u 0 (x) gives the error of

approximation 0( T). Plain calculations show that

( )

u(x, r) - u(x, 0)

u 1 x, 0 = -------

T

• au(x, 0) T a^2 x(x, 0) O(r2)
  
  
  • at + 2 at^2 + ·
  
  
  
  

From the governing differential equation it follows that

a^2 u(x, 0)

ax 2 +f(x,O)=Lu 0 (x)+f(x,O),

This is due to the fact that

In this line,

d^2 u 0 (x)

dx^2

u^1 ( ( ) a1t(x,O)^2

1 (x,0)- 2 r L1t 0 +f x,0) = at +O(r ).

Therefore, the difference initial condition Yt ( x, 0) = u 0 ( x), where

u 0 (x) = u(x) + ~ r(L U 0 + f(x, 0)),

approximates to second order in T the condition au( x' 0) I at = Uo ( x) on the
solution of problem (41).
In this case the condition u( x, 0) = u 0 ( x) and the boundary condi-
tions are approximated exactly. For instance, one of the schernes arising in
Section 1.2 is good enough for the difference approximation of the initial
equation. No doubt, we preassumed not only the existence and continuity
of the derivatives involved in the equation on the boundary of the domain
in view (at x = 0 or t = U), but also the existence and boundedness of
the third derivatives of a solution for raising the order of approximation of
boundary and initial conditions.