Молодой учёный

10 Математика
«Молодой учёный» . № 3 (50) . Март, 2013 г.
with a discrete function that satisfies the estimate
h
h
2
ξk -1 ≤ck ( - 1) h.
(29)
1
Because of Taylor series in x of r ( tk, x)
in the vicinity of point x i we get equality
xi+
12
k
r( tk, xi) = ∫ r( tk, x) dx h+ e
x
i
i-12
k
where ei
2
≤ ch 1
2
1 ∂ r
with c1 = max ( t , ) .
[0,1]
2 k x
24 x∈
∂x
(30)
Because of Theorem 1
k-1
xi+ 12 Ai+
12
r k = r k-1
x
k 1
i 12 A
-
- i-12
∫ ∫
( t , x) dx h ( t , x) dx h.
Instead of let use its piecewise linear periodical interpolant Then
k-1 Ai+ 12
r( t 1 1, ) k k x dx h -
A
- =
i-12 k-1
Ai+
12
k
r 1 int ( t 1,
)
k
A
k x dx h h
-
- + i
i-12
k
hi
≤ 2
k-1 i+ 12- k-1
i-12 1
2 =
x∈[0,1]
2
∂ r
2 k
∫ ∫
where chA ( A ) with c max ( t , x)
.
8 ∂x
Thus, we get equality
k-1
Ai+
12
k k
k i = k-1
A
int k -1
+ i + i
i-12
r( t , x ) ∫ r ( t , x) dx h h e .
(32)
For we use (21) and (28):
k-1 k-1
Ai 12 A
h + i+
12 h
k i = k-1 int k 1 k 1
A
- + - k 1
i-12 A
-
i-12
∫ ∫ (33)
r ( t , x ) r ( t , x) dx h ξ ( x) dx h
where values of are constructed by piecewise linear periodical interpolation.
Now let subtract (33) from (32), multiply its modulus by h , and sum for all i = 0, 1,...,n–1:
r( t ,) r ( t ,)
⎛
⎜h h
⎝
e h ξ ( x) dx
⎞
⎟.
⎠
Due to Theorem 3 last terms ( ,) ⋅- in brackets ( ,) ⋅ is ≤ evaluated ( + ) by + . Thus
∑ ∫ (34)
k ⋅-
h
k ⋅
h
≤
1
0≤≤ i n-1
k
i +
k
i +
k-1
Ai+
12
k-1
Ai-12
h
k-1
r tk h
r tk h
1
c1 c2 2
h
h
h
ξk -1
1
h
h
2 h
h
k k
1
1 2 k 1
1
r( t ,) ⋅-r ( t ,) ⋅ ≤ ( c + c ) h + ξ - .
(35)
Let put c = c 1 + c 2 then this inequality is transformed with the help (29):
h
r( t ,) ⋅-r ( t ,) ⋅ ≤ ( c + c ) kh
k k
h
1
1 2
that is equivalent to (27).
We can see that at last level we get inequality
h
h
2
( tm,) ( tm,) cT h .
1
2
r ⋅-r ⋅ ≤ t
(36)
In some sense we got a restriction on temporal meshsize t to get convergence. For example, to get first order of convergence,
it is enough to take
t = ch
with any constant c independent of t and h. But this restriction is not such strong up to constant as Courant–Friedrichs–
Lewy (CFL) condition:
(31)