Các pp Runge – Kutta hiện giải bài toán Cauchy cho phương trình vi phân thường
cuu duong than cong . co m
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Bài toán Cauchy
),
,
I
x X , 0
k
,
)
f x y x ( , 1 y C I R
y 0
y ' ( y x 0
cuu duong than cong . co m
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Phương trình tích phân
x
f
t y t dt ,
y x
y x 0
x 0
x k
1
f
t y t dt ,
y x k
y x k
1
cuu duong than cong . co m
x k
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,
)
• Euler forward (hiện) y y
hf x y ( n
n
n
1
n
• Euler backward (ẩn)
y
y
,
y
n
hf x n
n
n
1
1
1
• Công thức hình thang
cuu duong than cong . co m
y
y
y
,
,
f x y n
n
n
n
f x n
n
1
1
1
h 2
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R-K làm gì?
• Tính tích phân trong phương trình tích
phân qua s nấc trung gian
• Đảm bảo việc tính thông qua các nấc trung gian có hiệu quả giống như khai triển Taylor hàm y(x) đến bậc cao
cuu duong than cong . co m
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n
n
...
r k s s
y n
Công thức R-K tổng quát n r k 2 2
rk 1 1
y n 1
...
h y , n
i
n k i
n k i 1 1
n k ii i 1 1
hf x n
cuu duong than cong . co m
0,1
0, i
1
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R-K 1 nấc
s
1
n
y
y
n
r k 1 1
y
n
n
n 1 n k 1
2
hy
x
'
, y x
n
n
n
1
O h
cuu duong than cong . co m
hf x 1
y x r 1
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R-K 2 nấc
s
2
n rk 2 2
,
y 1 n n k 1
hf n
n k 11 1
n k 2
h y , 2 n
hf 2
' , x n
n y rk 1 1 n hf x y n n hf x n n h f k 2 n
n ' 2 k f O h ( ) 11 1 , y n
cuu duong than cong . co m
f
y x n
hf n
' f , x n
' . , y n n
y x 1 n
3 f O h
2 h 2
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;
1 1
R-K 2 nấc 1; r r 2 2 2
r 1
r 2
1;
;
0;
r 2
2
1 1
r 1
1 2 1 2
1 2 1 2
1
;
1 1
2
r 1
r 2
1 2
;
;
cuu duong than cong . co m
1 1
r 2
2
r 1
2 3
1 3
3 4
.....
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R-K 3 nấc
n
y n
rk 1 1
n r k 3 3
n r k 2 2
,
y n 1 n k 1
n
, h y n
k 11 1
2
n k 2
n
, h y n 3
k 21 1
n k 3
n k 22 2
hf x y n n hf x n hf x n
cuu duong than cong . co m
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R-K 3 nấc
n
n
' , x n
' , y n
2
2
f hf f hf 2 11
n 2
3
k
2 2
2 11
2 2 21
n
2 n
" , x n
'' , x y
'' , y n
O h
f f f f f h
2
n
n
h 2 h
2 3
n
' , x n
' , y n
'' xx
n 3
2
3
n
n
n
n
f hf f f 3 k 21 1 k 22 2 h 2 h 2 k
3
'' xy
'' yy
O h
h
2
f f k 22 2 k 21 1 k 21 1 k 22 2 h
' , x n
' y
n
n
y x n
1
n
3
cuu duong than cong . co m
4
hf f f f . y x n h 2
'' xx
'' xy
n
'' yy
2 n
' y
' x
'2 y
n
O h
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f f f f f f f f f h 6
1
r 1
r 2
r 3
r 2 3
r 3
2
1 2
21
r 3
r 2 11
22
1 2
2 r 3 3
2 r 2 2
1 6
1 2
1 2
r r 2 3 3 22
21
21
2
1 6
2 22
2 r 11 2
2 21
r 3
1 6
22 2
cuu duong than cong . co m
11 22
1 6 1 6
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R-K3 thường dùng
;
1;
2
;
;
1;
21
22
r 1
r 2
r 3
11
; 3
; 2
1 6
1 2
1 6
1 2
4
y
y
k
k
1
n k 1
n 2
n 3
n
n
2 3 1 6
,
y
n k 1
n
n
x
k
hf
, h y
n 2
n k 1
n
n
1 2
1 2
2
k
, h y
k
n 3
n 2
n k 1
n
n
hf x hf x
cuu duong than cong . co m
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R-K3 thường dùng (Heun)
;
0
;
0;
; 22
3
21
r 1
r 2
2
r 3
; 11
2 3
1 4
3 4
1 3
y
y
k 3
n 3
n k 1
n
n
1
1 4 ,
n
n k 1
k
hf
h y ,
x n
n
n 2
n k 1
1 3
1 3
hf x y n
cuu duong than cong . co m
k
hf
k
h y ,
x n
n
n 3
n 2
2 3
2 3
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R-K 4 thường dùng
2
2
k
k
k
y
y
n 4
n 2
n 3
n k 1
1
n
n
1 6 ,
n k 1
n
, h y
hf
k
n k 1
n 2
x n
n
1 2
1 2
, h y
hf
k
k
n 2
n 3
x n
n
1 2
cuu duong than cong . co m
, h y
k
k
n 3
n 4
n
hf x y n hf x n
1 2
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Bậc cao nhất của các công thức R_K s nấc
s
1
4
3
6
5
2
7
8
9
p
1
4
4
3
5
2
6
6
7
cuu duong than cong . co m
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Ví dụ mô hình hệ thú mồi
1
rn
x
'
ap
p
'
n K p anp
cuu duong than cong . co m
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