45
Finite Volume Methods for Hyperbolic Problems Zhiping Li LMAM and School of Mathematical Sciences Peking University

LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

Finite Volume Methods for Hyperbolic Problems

Zhiping Li

LMAM and School of Mathematical SciencesPeking University

Page 2: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 3: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 4: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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�3Ø­½ê�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�� ���Ø­½5. 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

Page 5: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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�Ø­½, ��KÑÑ��K�°Ý.

5 / 45

Page 6: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 7: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 8: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 9: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 10: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 11: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 12: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 13: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 14: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 15: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 16: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 17: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 18: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 19: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 20: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 21: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 22: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 23: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 24: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 25: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 26: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 27: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 28: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 29: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 30: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 31: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 32: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 33: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 34: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 35: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 36: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 37: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 38: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 39: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 40: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 41: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 42: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 43: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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

Page 44: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

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)).

44 / 45

Page 45: LMAM and School of Mathematical Sciences Peking University · Lecture 11: Chap. 15, x15.3.5 |{; Chap. 16, x16.1 ı‡5¯ð˘’§|˙k†N¨{ı‡5¯ð˘’§|iøflK˙‡5z’{||

��: 15.2, 16.1

Thank You!