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Finite Volume Methods for Hyperbolic Problems
Zhiping Li
LMAM and School of Mathematical SciencesPeking University
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
�5z�{�":9Ö��{—– �?E
��§ª�/�Ó, �ÑyªÑ�¥%DÕÅ�, =�3 p ¦�3 p-Å��ýλp < 0, Ó�3 p-Å�mýλp > 0, K�5z�{(�) Roe �{) �Ñ� p-Åmä)Ø÷v Lax �^�. ÏdI��·��?Ö, =¤¢�?E (Entropy Fixes).
����?Ö�{´¦) p-È©�¤½Â�ODE Ð�¯K(� (13.48), (13.44)), ¿- Q↓i−1/2 = q(0).
~X, ®�k�� p-x�ªÑ�¥%DÕÅ, K�±l qpl Ñu,÷ p-È©�é�÷v λp(q↓) = 0 �: q↓, -Fi−1/2 = f (q↓).
é�kü�Å�fY�§, þã�{´�1�. �§|�ê�õ�, K��Ø�y¢, =B�±¢y, ó�þ���.
2 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
Cq�?E�{ —— Harten-Hyman ���
�iù¯K�5zCq)� k-Å��!mý�G�©O�qkl = Qi−1 +
∑k−1p=1Wp, qkr = qkl +Wk . �λkl < 0 < λkr .
� k-Å�Å� (�5z�{Ý�1k�A��) � λk .
ò k-ÅWk ©)�©O±Å�λkl Ú λkr DÂ��1ÅWk
l = βWk Úm1ÅWkr = (1− β)Wk , β ∈ (0, 1).
dÅð5,©)c�Ïþ�ØC, =λkl Wkl + λkrWk
r = λkWk .
u´�β = λkr−λkλkr−λkl
. - (λk)− = βλkl , (λk)+ = (1− β)λkr .
�ª�òÅ�L«���A±∆Q =∑m
p=1(λp)±Wp.
('� p315 (15.10))
3 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
Harten-Hyman ����,�«)º: O\ê�Ê5
Roe �{�ê�Ïþ�±ÏLÙÝA��L«�
Fi−1/2 =1
2[f (Qi−1) + f (Qi )]− 1
2|Ai−1/2|(Qi − Qi−1)
=1
2[f (Qi−1) + f (Qi )]− 1
2
∑p
|λpi−1/2|Wpi−1/2.
ù�±À�3ؽê�Ïþ 12 [f (Qi−1) + f (Qi )] Ä:þ\
þ*ÑÏþ −12
∑p |λ
pi−1/2|W
pi−1/2, ½ê�Ê5.
��5z k-Å´ªÑ��� Lax--Å�, |λki−1/2| Ï~�©�Cu", ù«�Ôn-Å�±)º�Roe �{ k-Å�ê�Ê5�� ���ؽ5. ý)A�ªÑ�¥%DÕÅ.
3Harten-Hyman ���¥, (λk)+−(λk)−=(1−β)λkr +β|λkl |O� |λki−1/2|, ùk�/O\ k-Å�ê�Ê5.
4 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
ÏLO\ê�Ê5������{—— Harten ���
Harten �����{�\��§=ò |λpi−1/2| �� φδ(λpi−1/2),
Ù¥
φδ(λ) =
{|λ|, |λ| ≥ δ,λ2+δ2
2δ , |λ| < δ.
½ö�d/^ (λ)− , 12 [λ− φδ(λ)], (λ)+ , 1
2 [λ+ φδ(λ)] ?�
(λpi−1/2)− Ú (λpi−1/2)+.
`:: {ü, O�þ� (ØI�O� λkl = λk (f ′(qkl )), λk
r = λk (f ′(qkr ))).
":: δ I��¯KN�. ��KØU�Ñê�ؽ, ��KÑÑ��K�°Ý.
5 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
ÏLO\ê�Ê5������{—— LLF ���
,�«O\ê�Ê5��{´ò�§ª� LLF �{��§|�Roe �5z�{�(Ü.
�§ª� LLF �{ (12.12) í2��§|�/�
Fi−1/2 =1
2[f (Qi−1) + f (Qi )]− 1
2
∑p
αpi−1/2W
pi−1/2,
Ù¥ αpi−1/2 = max{|λpi−1|, |λ
pi |}, λ
pi−1, λpi ´ f ′(Qi−1), f ′(Qi ) �
1 p �A��.
ò (15.9) ¥�Ïþ f (Q↓i−1/2) ^±þ LLF �Fi−1/2 O�, ¿|^
Roe �5z�{L« f (Qi )− f (Qi−1) =∑m
p=1 λpi−1/2W
pi−1/2, �
A∓∆Qi−1/2 =1
2
∑p
(λpi−1/2 ∓ αpi−1/2)Wp
i−1/2.
6 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
LLF ����`":
`:: ùp^αpi−1/2 �O (15.54) ¥�φδ(λ), Ïd�Harten �{
aq, {ü�O�þ�.
":: ØشĪÑ�¥%DÕÅ, ¤k�ÅÑO\ê�Ê5.ØL3)1w?, du λpi−1/2 ≈ λ
pi−1 ≈ λ
pi , ?�c�«O¿Ø�.
5: ±þA«����{3��5�§ªàÏþ�/�Ñ�Ñ´E �ª, Ïd�±y²�Aê�)Âñu�f). é�§|vk�A(J.
7 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K��5z�{—— �5z�{�":9?Ö�{
3,AÏ�¹e�5z�{�U�Ñ�Ôn)
Ø�´Ø����¯K, ½½5�¯K. 3�AÏ�¹e�5z�{�U¬�Ñ�Ôn).
~X, iùÐ�� hl = hr = 1, ur = −ul = 1.8 £g = 1¤�fY�§iù¯K, e^Roe �5z�{¦), ���¥mG��Y�´K� (ê�(J�p327ã 15.3(b)).
éuNõkÔn�µ�¯K, �{���5�©�.
�éî.�§�,iù¯K, Ø�3����5z�{.
�±y²æ^iù¯Ký)�Godunov �{, ±9e¡ò�0��HLLE iù¯KCq¦)�{äk��5.
8 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5V.ÅðÆ�§|iù¯K�Cq¦)�{—— HLL Ú HLLE Cq¦)�{
iù¯K� HLL Ú HLLE Cq¦)�{
,�aiù¯KCq¦)��{Äu±eµe:
�Oiù¯K)¥��Ú���Å� s1i−1/2, s2
i−1/2.
�iù¯K�Cq)=¹Å�� s1i−1/2, s2
i−1/2 �ü�Å.
ùü�Å�rÝ�d���¥mG� Qi−1/2 L«�
W1i−1/2 = Qi−1/2 − Qi−1, W2
i−1/2 = Qi − Qi−1/2.
dùü�Å|¤�iù¯K�Cq)A÷vÅð5, =
s1i−1/2(Qi−1/2−Qi−1)+s2
i−1/2(Qi−Qi−1/2) = f (Qi )−f (Qi−1).
dd�: Qi−1/2 =f (Qi )− f (Qi−1)− s2
i−1/2Qi + s1i−1/2Qi−1
s1i−1/2 − s2
i−1/2
.
5: Ta�{¥ØÓ�{�«OÌ�´ s1i−1/2, s2
i−1/2 ��{ØÓ.
9 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5V.ÅðÆ�§|iù¯K�Cq¦)�{—— HLL Ú HLLE Cq¦)�{
iù¯K� HLL Ú HLLE Cq¦)�{
Harten, Lax Ú van Leer �@JÑ� HLL �{ÀJiù¯Ký)�¤k�U�A��Ý�e.Úþ.�� s1
i−1/2, s2i−1/2. 5¿,
�iù¯Ký)�ý½mý�-Å�, K s1i−1/2 �U�u�ý-
Å�Ý, s2i−1/2 K�U�umý-Å�Ý.
Einfeldt 3dÄ:þJÑHLLE �{, Ù��/ª�
s1i−1/2 = min
p(min(λpi−1, λ
pi−1/2)), s2
i−1/2 = maxp
(max(λpi , λpi−1/2)),
Ù¥λpj ´ f ′(Qj) �1 p�A��, λpi−1/2 ´Roe ²þÝ�
1 p�A��.
10 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5V.ÅðÆ�§|iù¯K�Cq¦)�{—— HLL Ú HLLE Cq¦)�{
HLLE �{�`":9U?�{
`:: ÃØiù¯Ký)��úÚ�¯�Å´-Å�´¥%DÕÅ, HLLE �{Ñk�Ð�%C5, ��ØI�����. ��±y²HLLE �{äk��5.
":: �iù¯Ký)dn�±þ�Å|¤, �¥m�ÅrÝ���, HLLE �{é¥m�Å�©EÇ�é��. ~X, éî.�§, HLLE �{é�>mä�ê�(J�©EÇ�é��.
U?�{: HLLEM �{^�¹�>mä&E�©¡�5¼ê�O~ê��¥mG�. HLLEC ��{K3Cq)¥Ú\1n�Å(¹ü�¥mG�).
11 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5V.ÅðÆ�§|�p©EÇ�{
��5V.ÅðÆ�§|�p©EÇ�{
��5�§|aq/�±|^Godunov �{£½Ù§Äuiù¯KCq¦)�{¤�)�E��5�§|�p©EÇ�ª:
Qn+1i = Qn
i −∆t
∆x(A−∆Qi+1/2+A+∆Qi−1/2)−∆t
∆x(Fi+1/2−Fi−1/2),
Ù¥ Fi−1/2 = 12
∑Mwp=1 |s
pi−1/2|(1− ∆t
∆x |spi−1/2|)W
pi−1/2. W p
i−1/2
´ p-ÅW pi−1/2 �,«��/ª (�§ 6.15).
éu-ÅÚ�>mä, W pi−1/2 = qpr − qpl ��5��Ó; éu¥
%DÕÅ, E�òÙÀ�TÅ�rÝ, d�Å�¿���ü��þ, ØL3�ª¥Ï~�±� sp �²þÅ� sp = 1
2 (λpl + λpr ).
12 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5V.ÅðÆ�§|�p©EÇ�{
��5V.ÅðÆ�§|�p©EÇ�{
5 1: ��ÞáÅÚp©EÇ?�Ü©�±^Ó���{��±^ØÓ��{���Å5�E. ~X, ��ÞáÅ A±∆Qi−1/2
�^iù¯K�ý)½HLLE �{½�½Ø�����Roe �5z�{�Ñ�Cq)�E, p©EÇ?�Ü©� Fi−1/2 �^�½Ø�����Roe �5z�{�Ñ�Å�E.
5 2: W pi−1/2 I�'� W p
i−1/2 � W pI−1/2 (� (6.61)) 3Ùþ�
ÝK��2ÀJ·��p©EÇ��ìφ(θ) (� (6.39(b)))û½.
~X, - θi−1/2 =W p
i−1/2·W p
I−1/2
W pi−1/2
·W pi−1/2
, W pi−1/2 = φ(θi−1/2)W p
i−1/2.
~: ã 15.5 Úã 15.6 ©OЫÚ'�ÄuRoe �{ÚHLLE�{�Godunov �{Ú�Ap©EÇ�{)î.�§Woodward-Colella �ñůK (� (15.65)) �ê�(J.
13 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
iù¯KCq¦)�{�,�«ÅDÂ/ª—— Ïþ�©�/ª
iù¯KCq¦)�{�Ïþ�©�/ª
c¡·�òG��©�¤±Å� spi−1/2 DÂ�Å, ¿½ÂÞá:
Qi −Qi−1 =∑Mw
p=1Wpi−1/2, A±∆Qi−1/2 =
∑Mwp=1(spi−1/2)±Wp
i−1/2.
,�«�{´òÏþ�©�¤±Å� spi−1/2 DÂ�Å, =
f (Qi )− f (Qi−1) =∑Mw
p=1Zpi−1/2,
,���|^Zpi−1/2 ½ÂÞáA
±∆Qi−1/2 =∑
p:spi−1/2
≷0Zpi−1/2,
¿�±½Âp©EÇ��Ïþ
Fi−1/2 =1
2
Mw∑p=1
sign(spi−1/2)(1− ∆t
∆x|spi−1/2|)Z
pi−1/2,
Ù¥ Zpi−1/2 ´Z
pi−1/2 �,«�� (aqu Wp
i−1/2).
14 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
iù¯KCq¦)�{�,�«ÅDÂ/ª—— Ïþ�©�/ª
iù¯KCq¦)�{�Ïþ�©�/ª�`:
51: � Ai−1/2 ��Roe ²þ�, Ïþ�©��G��©��Ñ
��{´���Ó�. �Ïþ�©��^u���� Ai−1/2, ~
X, �� Ai−1/2 = f ′( 12 (Qi−1 + Qi )). d�, Äu�5zÝ
Ai−1/2 �Ïþ�©���{�±�y÷v±eÄ�5�
1 �¢é�z;
2 � Qi−1, Qi → q �, Ai−1/2 → f ′(q);
3 ’ A−∆Qi−1/2 +A+∆Qi−1/2 = f (Qi )− f (Qi−1) (Åð5).
52: Ïþ�©�.�{(ÜÞáÅúª¦^´Åð.�{.
15 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
�{31w)�/U����°Ý�^�
31w)�/�{�ÛÜ�äØ�©Û
¯K: 31w)�/, ±þ=�{(Ã��ì�)U����°Ýº
�:: òê�)�ý)�Taylor Ðm�'�, y²�ªk��ÛÜ�äØ�. e�{�k��5½5, K�NØ�����.
éÅðÆ�§| qt +f (q)x =0, � q, f ¿©1w, K
qt = −f (q)x ,
qtt = −(f (q)x)t = −(f (q)t)x = −(f ′(q)qt)x = (f ′(q)f (q)x)x .
Ïd, k Taylor Ðmª
q(xi , tn+1) =[q −∆tf (q)x +
∆t2
2(f ′(q)f (q)x)x
]∣∣∣(xi ,tn)
+ O(∆t3).
16 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
�{31w)�/U����°Ý�^�
31w)�/�{�ÛÜ�äØ�©Û
,��¡, P Ai−1/2 = A(Qi−1,Qi ) �iù¯KCq)¤éA�Ïþ¼ê f (q) �Cq Jacobi Ý; P Zp
i−1/2, spi−1/2 �Ïþ�©
�� p-Å��A�Å�, §�÷v'X
f (Qi )− f (Qi−1) =m∑
p=1
Zpi−1/2, A±∆Qi−1/2 =
∑p:sp
i−1/2≷0
Zpi−1/2.
p©EÇ��Ïþ�
Fi−1/2 =1
2
m∑p=1
sign(spi−1/2)(1− ∆t
∆x|spi−1/2|)Z
pi−1/2,
ò§��\Ã��ì�p©EÇ�ª
Qn+1i = Qn
i −∆t
∆x(A−∆Qi+1/2+A+∆Qi−1/2)−∆t
∆x(Fi+1/2−Fi−1/2)
17 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
�{31w)�/U����°Ý�^�
31w)�/�{�ÛÜ�äØ�©Û
�n�
Qn+1i =Qn
i −∆t
2∆x
[ m∑p=1
Zpi−1/2+
m∑p=1
Zpi+1/2
]+
∆t2
2∆x2
[ m∑p=1
spi+1/2Zpi+1/2−
m∑p=1
spi−1/2Zpi−1/2
]=Qn
i −∆t
2∆x
[ m∑p=1
Zpi−1/2+
m∑p=1
Zpi+1/2
]+
∆t2
2∆x2
[Ai+1/2
m∑p=1
Zpi+1/2−Ai−1/2
m∑p=1
Zpi−1/2
]=Qn
i −∆t
2∆x
[f (Qi+1)−f (Qi−1)
]+
∆t2
2∆x2
[Ai+1/2(f (Qi+1)−f (Qi))−Ai−1/2(f (Qi)−f (Qi−1))
].
òý) q �\þª, �±wÑ��3/ªþùp�n�©OéAý)�Taylor Ðmª�cn�. e¡·�é1�Ú1n��äN©Û, ¿�Ñ���ª¤A÷v�^�.
18 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
�{31w)�/U����°Ý�^�
31w)�/�{�ÛÜ�äØ�©Û
d f (qi+1)−f (qi−1) = 2f (qi )x∆x +O(∆x3) ÚO(∆x)=O(∆t) �
∆t
2∆x
[f (Qi+1)− f (Qi−1)
]= ∆tf (qi )x + O(∆t3).
=3��°Ý�¿Âe�ª�1���ý)Taylor Ðmª�1����.
�©Û1n�, P
A(q(x), q(x + ∆x))f (q(x + ∆x))− f (q(x))
∆x= f ′(q(x + ∆x/2))f (q(x + ∆x/2))x + E2(x ,∆x).
dTaylor Ðm� f ′(qi+1/2)f (qi+1/2)x − f ′(qi−1/2)f (qi−1/2)x
= (f ′(qi )f (qi )x)x∆x + O(∆x3).
19 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
�{31w)�/U����°Ý�^�
31w)�/�{k��ÛÜ�äØ��^�
Ïd, XJ E2(x ,∆x) ÷v^�
F : E2(x ,∆x) = O(∆x),E2(x + ∆x ,∆x)− E2(x ,∆x)
∆x= O(∆x),
K��°Ý�¿Âe�ª�1n��ý)Taylor Ðmª�1n���.
±þ©ÛL², �iù¯KCq�{�Ý A(q(x), q(x + ∆x))÷v^�F �, Ã��ì�p©EÇ�ªk��ÛÜ�äØ�.
´y: eiù¯KCq Jacobi Ý A(q(x), q(x + ∆x)) ÷v^�
� : A(q(x), q(x + ∆x)) = f ′(q(x + ∆x/2)) + E (x ,∆x),
E (x ,∆x) = O(∆x),E (x + ∆x ,∆x)− E (x ,∆x)
∆x= O(∆x),
KF ¤á, ÏdÃ��ì�p©EÇ�ªk��ÛÜ�äØ�.
20 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
31w)�/k��°Ý��{
31w)�/k��°Ý��5z�{�~
AO/, � A(q(x), q(x + ∆x)) = f ′(q(x + ∆x/2)) + O(∆x2) �,�A�Ã��ìp©EÇ�ªk��ÛÜ�äØ�.
~X, � Qi−1/2 = 12 (Qi−1 + Qi ) ½� Qi−1/2 � Qi−1 Ú Qi �
Roe ²þ, - A(Qi−1,Qi ) = f ′(Qi−1/2), Kk
A(q(x), q(x + ∆x)) = f ′(q(x + ∆x/2)) + O(∆x2).
Ïd, �A�Ã��ìp©EÇ�ªk��ÛÜ�äØ�.
5 1: ^�F '^�� (¢�f, Ï�cö��¦A 3��AÏ��þkÐ�%C5�, �ö%�¦��Ý%C�Ð.
5 2: �Ïþ/� f (x , q) �, ±þ©Û�¤á.
21 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
31w)�/k��°Ý��{
üÚ Lax-Wendroff �{—– ÃICq Jacobi Ý����{
ÏLüÚ Lax-Wendroff �{, ÃIO�Cq Jacobi ÝÚiù¯K)�Cq(�,��±3)1w?����°Ý.
~ 1: Richtmyer �{: Qn+1i = Qn
i −∆t∆x (f (Q
n+1/2i+1/2 )− f (Q
n+1/2i−1/2 )),
Qn+1/2i−1/2 = 1
2 (Qni−1/2 + Qn
i )− ∆t2∆x [f (Qn
i )− f (Qni−1)]. (� §4.7)
~ 2: MacCormack �{: Qn+1i = 1
2 (Q∗i + Q∗∗i ), Ù¥
Q∗i = Qni −
∆t
∆x(f (Qn
i+1)− f (Qni )), (�m)�c�©,
Q∗∗i = Q∗i −∆t
∆x(f (Q∗i )− f (Q∗i−1)), (�m)���©.
�c���©�^S��±Øä��±����é¡��{.
22 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
31w)�/k��°Ý��{
üÚ Lax-Wendroff �{�`":
`:: �;�CqO� Jacobi Ý f ′(·), ØI�¦)½Cq¦)iù¯K, Ó�3)1w?qk��°Ý.
":: N´�)ê���, ��I�\<óÊ5. �<óÊ5¿��\3iù¯K)ý�I�\Ê5�Åþ, Ï q¬ÚåLr�*Ñ£½ÑѤ.
23 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K�Cq¦)ì—— Ïþ�þ©�{
Ïþ�þ©�{�Ä�µe
��òiù¯KCq¦)ïá3Ïþ�þ©��Ä:þ:
PAi = f ′(Qi ), λpi , rpi , i = 1. · · · ,m, �ÙA���A��þ.
òÏþ�þ f (Qi) ^TA��þ|�5LÑ f (Qi )=m∑
p=1
φpi rpi .
½Â φp(−)i =
{φpi , λpi < 0,
0, λpi ≥ 0.φp(+)i =
{φpi , λpi > 0,
0, λpi ≤ 0.
½Â f(±)i =
m∑p=1
φp(±)i rpi , KkÏþ�þ©� f (Qi) = f
(+)i +f
(−)i .
51: AO/, é÷v f (αq) = αf (q) (½ f (q) = f ′(q)q)��gàgÏþ¼êk
A+i Qi + A−i Qi = AiQi = f (Qi ) = f
(+)i +f
(−)i .
52: éªÑ�6, ù«©�Ø1w, )�)�¬kÂñ5(J, I�U?.
24 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
��5ÅðÆ�§|iù¯K�Cq¦)ì—— Ïþ�þ©�{
Ïþ�þ©�{�Ä�µe
½Â.¡ xi−1/2 ?�ê�ÏþFi−1/2 = f(+)i−1 + f
(−)i , ±9Þ
áÅA−∆Qi−1/2 = f(−)i − f
(−)i−1 , A+∆Qi−1/2 = f
(+)i − f
(+)i−1 .
PQi =m∑
p=1
ωpi r
pi , Kk Qi − Qi−1 =
m∑p=1
(ωpi r
pi − ω
pi−1r
pi−1).
�±½ÂWpi−1/2 = ωp
i rpi − ω
pi−1r
pi−1, spi−1/2 = 1
2 (λpi−1 + λpi ).
��±½ÂZpi−1/2 = φpi r
pi − φ
pi−1r
pi−1.
�d, ·�®²ÄuÏþ�þ©�O�Ñ�EGodunov �ª(� (15.2), (15.5)), Ú�Ep©EÇ�ª (� (15.62), (15.63) ½(15.71)) ¤I��¤kþ.
5: Ïþ�þ©�{����lÑENO 9Runge-Kutta �{(ܼ�p�°Ý.
25 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
'u��5ÅðÆ�§|�{�½5�Âñ5
�§|�oC�
oC�ØOTVD ÚoC�k.TVB 3��5ÅðÆ�§ª�{�½5�Âñ5©Û¥å�'�5��^.
��'�{¿Ø·^u����5ÅðÆ�§|. �Ï´��5ÅðÆ�§|�)����¿�oC�ØO (TVD) �,$�Ø÷vTV (Qn+1) ≤ (1 + α∆t)TV (Qn).
��~f, �ÄiùÐ�� hl = hr , ul = −ur > 0 �fY�§iù¯K (� p.266-267). Ð�¥, h �C�� 0, hu �C�� 2hlul . é?¿� t > 0, d um = 0, hu �C�E� 2hlul .
� h �C� 2(hm − hl)>2√
hlg ul (�(13.20) ¿|^ hm > hl ). -Щ
Äþ hlul = c2g �~þ, K�A)�oC� 2√
hlg ul = 2c
√ul
�±?¿�.
26 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
'u��5ÅðÆ�§|�{�½5�Âñ5
��5VÅðÆ�§|)��3��5, ê��{�½5ÚÂñ5
5 1: 8c, é,AÏ��5ÅðÆ�§|, �±y²Ù)±9Ù Godunov �{�TVB 5�, l y²ê�)�½5ÚÂñ5 (� p.345).
5 2: 3�½^�e, é,��5VÅðÆ�§|, �±y²Glimm �ÅÀ�{�Âñ5§l y²)��35.
5 3: é,��5VÅðÆ�§|, 3�½^�e, �±ÏLÖ��;�{, front tracking �{Ú�+�{� (� p.345), y²)��35, k�¹e��±y²��5.
27 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
-ÅO�¥�ê���
dåÚØ�Úå�-ÅO�¥�ê���
�Щ^��Ñ� p-x-Å. ���mÚ��, eT-ÅDÂålî��u���mÚ�, Kd²þ�O�ò�Ñ��#�G�.
XJ�A�Hugoniot :8���, e��mÚ�ê�)ò�¹Nõ�Å, Ø´��ü�ü�þ�ü� p-x-Å.
ùÒ´åÚØ�Úå�ê���. duê�Ê5, ù��)���Ï~¿ØwÍ. Ø�ê�Ê5��.
ã 15.8 w«æ^\minmod ��ìp©EÇ���Godunov �{O�î.�§��� 3- -Å (s = 10.9636)��ê�(J. l¥�±*�dåÚØ�Úåê���.
28 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��5ÅðÆ�§|�k�NÈ{
-ÅO�¥�ê���
$�-ÅO�¥�ê���
éÅ��Cu"�-Å, Godunov �ªÚ Roe �ª���{ê�Ê5��, cÙ´I�Nõ�mÚâUªL���mÚ��, -ÅO��zÚѬ�)#���, ù��¬p�U\, Ïd, ��ò¬'�²w.
ã 15.9 w«æ^\minmod ��ìp©EÇ���Godunov �{O�î.�§��� 3- -Å (s ≈ 0.4646) ��ê�(J. �±wÑdåÚØ�Úåê���duÅ��$ �?�Ú\r.
ê�Ê5���¬�5NõÙ§ê�¯K (� p.348). é$�-Å\þ·�<óÊ5éuO�½5´7��. 3p��/, cÙXd.
29 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
�.¯K —— ü�6�Buckley-Leverett �§
à5£ý���5¤3V.¯K), cÙ´iù¯K)�(�©Û¥å��©���^. vkà5, )�(�Ò¬u)¢�5�UC. ·�òÏL±e�.¯K�«Ún)Ù¥�Cz.
��ü�õ�0�+�6�.¯K: �+�d��¡õ�0��¤, 6NØ�Ø, ��¡þ6���� 1, YÚh��Ý�~ê.
q(x , t): t-�� x-?Y��ÚÝ(ü NÈ6N¥Y�NÈ),
1− q(x , t): t-��, x-?h��ÚÝ. 0 ≤ q ≤ 1.
Y��ÚÝÏþ¼ê f (q) =q2
q2 + a(1− q)2. 0 < a < 1 �~ê.
h��ÚÝÏþ¼ê fo(q) =a(1− q)2
q2 + a(1− q)2. f (q) + fo(q) ≡ 1.
f ′(q) =2aq(1− q)
[q2 + a(1− q)2]2. (a = 1
2� f , fo Ú f ′ �p.352 ã16.1).
30 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
Buckley-Leverett �§�iù¯K
äkþãÏþ¼ê�ÅðÆ�§ª qt + f (q)x = 0 ¡�Buckley-Leverett �§, §´�gæh{z�.�§.
éu��5ÅðÆ�§ª£�)�àÏþ�/¤iù¯K�Ôn)´�q5), A��{Ú�¡È{KE,k�, -Å7L÷vOleinik �^�.
Definition
� q(x , t) ´�����ÅðÆ�§ª�f), XJ q(x , t) �¤kmäÑ÷v
f (q)− f (q∗l )
q − q∗l≥ s ≥ f (q)− f (q∗r )
q − q∗r, u q∗l , q∗r �m� ∀q,
Ù¥ q∗l , q∗r ©O´��mä�müý�G��, K¡ q(x , t) ÷vOlinik �^�. £5µd� q(x , t) 7´TÅðÆ�§ª�Ê5��).¤
31 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
Buckley-Leverett �§�iù¯K
�Ä�gæh{z�.Buckley-Leverett �§ qt + f (q)x = 0,
Ù¥ f (q) =q2
q2 + a(1− q)2, 0 < a < 1 �~ê.
�ÄiùÐ�� ql = 1, qr = 0 �iù¯K�¦).
éÐ���6Ä, - q0ε (x) =
1, x < −ε,ε−x2ε , −ε ≤ x ≤ ε,
0, x > ε.
.
u´dA��{, l x ∈ [−ε, ε] Ñu÷�ÚÝ� q0ε (x) �A
��3 t ����� �´ x + f ′(q0ε (x))t.
32 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
^A��{Ú�¡È{K¦)B-L �§�iù¯K
- ε→ 0, �l 0 Ñu÷�ÚÝ� q ∈ [0, 1] �A��3 t ����� �´ f ′(q)t (�p.353ã16.2(a), '�ã16.1(b)).
w,A��{�Ñiù¯K�kn���õ�).
d�¡È{K, ·��±é�� q∗ ∈ (0, 1), ¦�Ú\-Å�f)�A��{�Ñ�f)¤�¡È�� (�ã 16.2(b)). ù¿�X: Y�oþ=Y�ÚÝ×õ��¡È×+��Ý=~þ,=Y�oþÅð(½Y�o�þÅð).
dR-H a�mä^�, -Å��Ý s = f (q∗)−f (qr )q∗−qr = f (q∗)
q∗ .
q st = f ′(q∗)t, ¤±, s = f ′(q∗) = f (q∗)q∗ .
33 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
A��+�¡È{K�Ñ�B-L �§iù¯K)�(�
ã 16.2(c) w«�´ x-t ²¡þ�d�¡È{K�Ñ�)�A��©Ù(�. �±wÑ, 3 0 ≤ x/t ≤ s = f ′(q∗) �, )´��¥%DÕÅ, x/t = s = f ′(q∗) ?k�rmä, =-Å.
¯¢þ, dã 16.2(b) �, ±þB-L �§iù¯K�)�
q(x , t) =
ql = 1, x/t ≤ 0 = f ′(ql),
q(x/t), x/t ∈ (f ′(ql), f′(q∗)),
qr = 0, x/t ≥ f ′(q∗)
Ù¥ q(x/t) ∈ (q∗, ql) ÷v x/t = f ′(q(x/t)).
5 1: 5¿, é±þ¯K, f ′(q) 3 (q∗, 1) þî�üN~.
5 2: Ôn)º: 3 t∗ = x∗/f ′(q∗) �c, l u x∗ > 0 �³¥6Ñ�Ñ´h. ��KY�'~ÅìO\, �[�¹h.
34 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
A��+�¡È{K�Ñ�B-L �§iù¯K)�(�
Ón��, �ql = 0, qr = 1 �, B-L �§iù¯K�)�
q(x , t) =
ql = 0, x/t ≤ 0 = f ′(ql),
q(x/t), x/t ∈ (f ′(ql), f′(q∗)),
qr = 1, x/t ≥ f ′(q∗)
Ù¥ q(x/t) ∈ (ql , q∗) ÷v x/t = f ′(q(x/t)). ùp, f ′(q)
3 (0, q∗) þî�üNO.
±þB-L �§iù¯K�)ÑÓ��¹��¥%DÕÅÚ��-Å (iù¯K�)�¹��deZ¥%DÕÅÚ-Å|¤�EÜÅ´�àÏþ��A:). 3àÏþ�, iù¯K�)�o´��-Å, �o´��¥%DÕÅ.
35 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
��£¹�àÏþ¤ÅðÆ�§ª-Å��^�
5¿, B-L �§iù¯K�)¤�¹�-ÅØ´ Lax--Å, =Ø÷vλl > s > λr . éu��£¹�àÏþ¤�ÅðÆ�§ª, Oleinik Ú\±eOleinik �^�:
Definition
��ÅðÆ�§ªÊ5��(f)) q(x , t) �¤kmäÑ÷v
f (q)− f (q∗l )
q − q∗l≥ s ≥ f (q)− f (q∗r )
q − q∗r, u q∗l , q∗r �m� ∀q,
Ù¥ q∗l , q∗r ©O´��mä�müý�G��.
N´�y, éukî�àÏþ�ÅðÆ�§ª, ÷v Oleinik�^��-Å7,´ Lax--Å.
36 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
äk�àÏþ¼ê���5VÅðÆ�§—— �.¯K: ü�6�Buckley-Leverett �§
B-L �§iù¯K�/A��+�¡È{K)0÷vOleinik �^�
1 ± ql = 1, qr = 0 �iù¯K�~, d� s = f ′(q∗) = f (q∗)q∗ .
2 3«m (qr , q∗) þ f (q) á3L: (qr , f (qr )) Ú: (q∗, f (q∗))
��� (ÙT� f �L (q∗, f (q∗)) ���)e� (�ã 16.3).
3 d�/A��+�¡È{K)0�-Å�müý�G�©O´ q∗l =q∗, q∗r =qr =0. u´d (2)�Ù÷vOleinik �^�.
4 �Ð�� ql = 0, qr = 1 �, k s = f ′(q∗) = f (q∗)−f (qr )q∗−qr , �
3«m (q∗, qr ) þ f (q) uL: (q∗, f (q∗)) Ú: (qr , f (qr ))��� (ÙT� f �L (q∗, f (q∗)) ���)þ�.
5 d�/A��+�¡È{K)0�-Å�müý�G�©O´ q∗l =q∗, q∗r =qr =1. u´d (4)�Ù÷vOleinik �^�.
37 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àVÅðÆ�§ªiù¯K�¦)�{ —— à�{
¦)�àVÅðÆ�§ªiù¯K�à�{
ɱþBuckley-Leverett �§iù¯K)�5�?Ø�éu, é����àVÅðÆ�§ªiù¯K, �±�Ä^à�{¦).ù´�«|^ f �ã�E5�¦iù¯K�)��{.
� qr <ql �, �8Ü {(q, y) : qr ≤q≤ql , y ≤ f (q)} �à�.
Tà��þ>.��deZ��ãÚeZ f (q) ��ã�Oë� ¤, 3z��ãþ f ′(·) �üN~¼ê;
Ù¥z���ãL«��ë�T��ãüàG��-Å,T��ã��Ç s =�-Å�Ý;
Ù¥z� f (q) ��ãKL«��ë�T�ãüàG��¥%DÕÅ, )3T�ãþ÷v x/t = f ′(q(x/t)).
ã 16.3 �ÑB-L �§, ql = 1, qr = 0 �à�)«¿ã.
38 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àVÅðÆ�§ªiù¯K�¦)�{ —— à�{
¦)�àVÅðÆ�§ªiù¯K�à�{
� qr >ql �, �8Ü {(q, y) : ql≤q≤qr , y ≥ f (q)} �à�.
Tà��e>.��deZ��ãÚeZ f (q) ��ã�Oë� ¤, 3z��ãþ f ′(·) �üNO¼ê;
Ù¥z���ã½�ã©OL«��ë�T�ãüàG��-Ž¥%DÕÅ. ��ã��Ç s ��A-Å�Ý, 3�ãþ)÷v x/t = f ′(q(x/t)).
ã 16.4(a) �Ñ f (q) = sin(q), ql = π4 , qr = 15π
4 �à�)«¿ã. Ù¥ q1 ≈ 4.2316 d π
4 < q1 < q2 = 3π2 ÚR-H ^�
sin(q1)−sin(π/4)q1−π/4 = cos(q1) �Ñ; q3 = 7π
2 . dd�, �A�)�
EÜÅdë� ql � q1, q2 � q3 �ü�-Å, ±9ë� q1 �q2, q3 � qr �ü�¥%DÕÅ|¤.
ã 16.4(b) �Ñ/A��+�¡È{K)0«¿ã.
39 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àVÅðÆ�§ªiù¯K�¦)�{ —— à�{
VÅðÆ�§ªiù¯K�à�{�Ñ�f)´�)
ql > qr �/: df)��E�, e q < q ´�ã-Å��müýG�, K {(q, f (q)) : q ∈ (q, q)} 7î�á38Ü {(q, y) : qr ≤q≤ql , y ≤ f (q)} �à�S. Ïd, Tã-Å7÷v Oleinik �^�. 3z� f (q) ��ãþ, f ′(q) 7´ q �üN~¼ê, Ïd�X q �~�, A��ÅìÑm.
ql < qr �/: Ón, e q < q ´�ã-Å��müýG�, K{(q, f (q)) : q ∈ (q, q)} 7î�á38Ü {(q, y) : qr ≤q≤ql , y ≥f (q)} �à�S. Ïd, Tã-Å7÷v Oleinik �^�. 3z� f (q) ��ãþ, f ′(q) 7´ q �üNO¼ê, Ïd�X q �O�, A��ÅìÑm.
Ïd, 3?Û�¹e, dà�{�E�f)Ñ´�).
40 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àVÅðÆ�§ªiù¯K�¦)�{ —— Osher )
�àVÅðÆ�§ªiù¯K�Osher )
�âà�{Ú¼êà��O��{, Osher �Ñ���àVÅðÆ�§ªiù¯K�q5�) q(x , t) = q(x/t) ���{üL�ª. �iùÐ�� ql , qr , - ξ = x/t, ½Â
G (ξ) =
{minql≤q≤qr [f (q)− ξq], ql ≤ qr ,
maxqr≤q≤ql [f (q)− ξq], qr ≤ ql .
K f (q(ξ))− ξq(ξ) = G (ξ), =
q(ξ) =
{argminql≤q≤qr [f (q)− ξq], ql ≤ qr ,
argmaxqr≤q≤ql [f (q)− ξq], qr ≤ ql .
5 1: eé,��½� ξ, ÷vþª� q �Ø��, K�)¥k���Ý� ξ �-Å, Ùüý�G�´÷vþªG�8�þe..
5 2: - f ∗(ξ) = supq(ξq − f (q)), f ∗∗(q) = supξ(qξ − f ∗(ξ)). Kf ∗∗(q) ´ f (q) �à�, = f ∗∗ �þã´ f þã�eà�.
41 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àVÅðÆ�§ªiù¯K�¦)�{ —— Osher )
Osher )�5�
5 3: é?�� q0, ·��±ò±þ½Â¥� f (q)− ξq �¤[f (q)− f (q0)]− ξ[q− q0], ùجUC(Ø. �C�ª�R-H ^��'X%kÏu·�n)(Ø.
5 4: éuB-L �§, ± ql = 1, qr = 0 �~, ´yà�{�Ñ�)÷v f (q(ξ))− ξq(ξ) = G (ξ). (SK: �Ñ��Ð��y².)
5 5: é'Xª f (q(ξ))− ξq(ξ) = G (ξ) 'u ξ ¦��
[f ′(q(ξ))− ξ]q′(ξ)− q(ξ) = G ′(ξ).
du q(ξ) ´�q5), Ïdðk [f ′(q(ξ))− ξ]q′(ξ) ≡ 0, u´�iù¯K�)���L�ª:
q(ξ) = −G ′(ξ).
42 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àÏþÅðÆ�§ªk�NÈ{
�àÏþÅðÆ�§ªk�NÈ{
3 f (q(ξ))− ξq(ξ) = G (ξ) ¥- ξ = 0, � ('�àÏþ�� (12.4))
f (q↓(ql , qr )) = f (q(0)) = G (0) =
{minql≤q≤qr f (q), ql ≤ qr ,
maxqr≤q≤ql f (q), qr ≤ ql .
-Fi−1/2 = f (q↓(Qi−1,Qi )), ·�Ò�±A^Godunov �{.
Ó�, -Q↓i−1/2 = q↓(Qi−1,Qi ), ,�½ÂÞá ('� (12.6))
A+∆Qi−1/2 = f (Qi )−f (Q↓i−1/2), A−∆Qi−1/2 = f (Q↓i−1/2)−f (Qi−1).
��Ep©EÇ�ª¤I�Å�Å�. ²�L², ���^��ű9�A�R-H mä^�½Â�Å�Òv:
Wi−1/2 = Qi − Qi−1, si−1/2 =f (Qi )− f (Qi−1)
Qi − Qi−1.
43 / 45
Lecture 11: Chap. 15, §15.3.5 —–; Chap. 16, §16.1
��²;���5V.¯K
�àÏþÅðÆ�§ªk�NÈ{
�àÏþÅðÆ�§ªk�NÈ{
5 1: ��±^Å�Å�½ÂÞá A±∆Qi−1/2 = s±i−1/2Wi−1/2.
5 2: �àÏþ�aq, XJf)¥�¹ªÑ�¥%DÕÅ, K7L����� (aqu (12.9)), ½öO\<óÊ5 (�§12.5 LL ÚLLF �{).
5 3: Courant ê= maxq
∣∣∣∆t
∆xf ′(q)
∣∣∣, Ù¥ q �Hf)�����
«� (à�), Ø==´f)���@G��. ¯¢þ, � f �à�f)k-Å�, cö (�«Oå�¡Ù�/ýCourant ê0)��î��u�ö (/�Courant ê0). ��AO�Ñ�´�mÚ�=÷v� Courant ê< 1 �, ê�)E�UÂñ.
ã 16.5(a) Ú (b) ©Ow«3O�iùÐ�� ql = 1, qr = 0 �B-L �§iù¯K�)�, �ýCourant ê< 1 Ú> 1 ���ê�(J. �ýCourant ê> 1 �, ê�)Âñ��´��-Å�Ý ú�Ø÷v Olinik �^��f)(�p.357 ã 16.5(b)).
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��: 15.2, 16.1
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