偏微分方程计算初步
偏微分方程计算初步
基本张量表达
在计算流体力学领域数学符号是理解和导出公式的关键部分。不管是人工推导还是计算机执行，在计算偏微分方程之前都需要掌握一些数学计算符号。本文中主要介绍一些必要的数学符号。通常一种场，例如标量，矢量以及一些经典的已知矩阵都可以用张量代表。因此，数值模拟可以说就是在处理n阶的张量 T n T^n Tn。为了更加清楚的理解这一点，我们做出如下定义：
0阶张量 T 0 T^0 T0:= C C C
一阶张量 T 1 T^1 T1:= a ⃗ \vec a a
二阶张量 T 2 T^2 T2:=3*3的矩阵，张量T
三阶张量 T 3 T^3 T3:= T i j k T_{ijk} Tijk
基本导数定义：
两个函数相加的导数等于函数导数的和：
∂ ( f 1 + f 2 ) ∂ t = ∂ f 1 ∂ t + ∂ f 2 ∂ t \frac{\partial{(f_1+f_2)}}{\partial{t}}=\frac{\partial{f_1}}{\partial{t}}+\frac{\partial{f_2}}{\partial{t}} ∂t∂(f1+f2)=∂t∂f1+∂t∂f2
∂ ( f 1 − f 2 ) ∂ t = ∂ f 1 ∂ t − ∂ f 2 ∂ t \frac{\partial{(f_1-f_2)}}{\partial{t}}=\frac{\partial{f_1}}{\partial{t}}-\frac{\partial{f_2}}{\partial{t}} ∂t∂(f1−f2)=∂t∂f1−∂t∂f2
两个导数相乘的导数等于分别对两个函数单独求导的和：
∂ ( f 1 f 2 ) ∂ t = f 1 ∂ f 2 ∂ t + f 2 ∂ f 1 ∂ t \frac{\partial{(f_1 f_2)}}{\partial{t}}=f_1\frac{\partial{ f_2}}{\partial{t}}+ f_2\frac{\partial{f_1}}{\partial{t}} ∂t∂(f1f2)=f1∂t∂f2+f2∂t∂f1
∂ ( f 1 f 2 f 3 ) ∂ t = f 1 ∂ f 2 f 3 ∂ t + f 2 f 3 ∂ f 1 ∂ t = f 1 f 2 ∂ f 3 ∂ t + f 1 f 3 ∂ f 2 ∂ t + f 2 f 3 ∂ f 1 ∂ t \frac{\partial{(f_1 f_2 f_3)}}{\partial{t}}=f_1\frac{\partial{f_2 f_3}}{\partial{t}}+f_2 f_3\frac{\partial{f_1}}{\partial{t}}=f_1f_2\frac{\partial{ f_3}}{\partial{t}}+f_1 f_3\frac{\partial{f_2}}{\partial{t}}+f_2 f_3\frac{\partial{f_1}}{\partial{t}} ∂t∂(f1f2f3)=f1∂t∂f2f3+f2f3∂t∂f1=f1f2∂t∂f3+f1f3∂t∂f2+f2f3∂t∂f1
一个常数C与函数的乘的导数：
∂ ( C f 1 f 2 ) ∂ t = C f 1 ∂ f 2 ∂ t + C f 2 ∂ f 1 ∂ t \frac{\partial{(Cf_1 f_2)}}{\partial{t}}=Cf_1\frac{\partial{ f_2}}{\partial{t}}+C f_2\frac{\partial{f_1}}{\partial{t}} ∂t∂(Cf1f2)=Cf1∂t∂f2+Cf2∂t∂f1
爱因斯坦求和约定
爱因斯坦求和约定就是省略求和符号。在流体动力学中，符号系统多种多样。最长的也是最简明的是笛卡尔形式。这种形式的表示可以使用爱因斯坦求和约定写成更为简单的形式。设一个函数在笛卡尔坐标下可以表示为如下所示的形式。
∂ f ∂ x + ∂ f ∂ y + ∂ f ∂ z = ∑ i ∂ f ∂ x i = ∂ f ∂ x i {\frac{\partial{f}}{\partial{x}}}+{\frac{\partial{f}}{\partial{y}}}+{\frac{\partial{f}}{\partial{z}}}=\sum_i {\frac{\partial{f}}{\partial{x_i}}}={\frac{\partial{f}}{\partial{x_i}}} ∂x∂f+∂y∂f+∂z∂f=i∑∂xi∂f=∂xi∂f
以上简单的式子并不能很好的体现爱因斯坦求和约定表示方法的优越性。再例如如下所示的式子：
∂ f x f x ∂ x + ∂ f y f x ∂ y + ∂ f z f x ∂ z + ∂ f x f y ∂ x + ∂ f y f y ∂ y + ∂ f z f y ∂ z + ∂ f x f z ∂ x + ∂ f y f z ∂ y + ∂ f z f z ∂ z = ∑ i ∑ j ∂ f j f i ∂ x j = ∂ f j f i ∂ x j {\frac{\partial{f_xf_x}}{\partial{x}}}+{\frac{\partial{f_yf_x}}{\partial{y}}}+{\frac{\partial{f_zf_x}}{\partial{z}}}+ {\frac{\partial{f_xf_y}}{\partial{x}}}+ {\frac{\partial{f_yf_y}}{\partial{y}}}+{\frac{\partial{f_zf_y}}{\partial{z}}}+{\frac{\partial{f_xf_z}}{\partial{x}}}+ {\frac{\partial{f_yf_z}}{\partial{y}}}+{\frac{\partial{f_zf_z}}{\partial{z}}}\\=\sum_i \sum_j {\frac{\partial{f_jf_i}}{\partial{x_j}}}=\frac{\partial{f_jf_i}}{\partial{x_j}} ∂x∂fxfx+∂y∂fyfx+∂z∂fzfx+∂x∂fxfy+∂y∂fyfy+∂z∂fzfy+∂x∂fxfz+∂y∂fyfz+∂z∂fzfz=i∑j∑∂xj∂fjfi=∂xj∂fjfi
将式子在笛卡尔坐标系中展开：
∂ ρ u i t + ∂ ρ u i u j x j = ∂ ρ u t + ∂ ρ v t + ∂ ρ w t + ∂ ρ u u x + ∂ ρ u v y + ∂ ρ u w z + ∂ ρ v u x + ∂ ρ v v y + ∂ ρ v w z + ∂ ρ w u x + ∂ ρ w v y + ∂ ρ w w z \frac{\partial{\rho u_i}}{t}+\frac{\partial{\rho u_i u_j}}{x_j}=\frac{\partial{\rho u}}{t}+\frac{\partial{\rho v}}{t}+\frac{\partial{\rho w}}{t}+\frac{\partial{\rho u u}}{x}\\+\frac{\partial{\rho u v}}{y}+\frac{\partial{\rho u w}}{z}+\frac{\partial{\rho v u}}{x}+\frac{\partial{\rho v v}}{y}+\frac{\partial{\rho v w}}{z}+\frac{\partial{\rho w u}}{x}+\frac{\partial{\rho w v}}{y}+\frac{\partial{\rho w w}}{z} t∂ρui+xj∂ρuiuj=t∂ρu+t∂ρv+t∂ρw+x∂ρuu+y∂ρuv+z∂ρuw+x∂ρvu+y∂ρvv+z∂ρvw+x∂ρwu+y∂ρwv+z∂ρww
张量数学计算
使用矢量替代爱因斯坦求和约定是一种常用的公式处理方法。例如如下几个矢量：
a ⃗ = ( a x a y a z ) = ( a 1 a 2 a 3 ) \vec a=\begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix}=\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} a=⎝⎛axayaz⎠⎞=⎝⎛a1a2a3⎠⎞
b ⃗ = ( b x b y b z ) = ( b 1 b 2 b 3 ) \vec b=\begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix}=\begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} b=⎝⎛bxbybz⎠⎞=⎝⎛b1b2b3⎠⎞
T = [ T x x T x y T x z T y x T y y T y z T z x T z y T z z ] = [ T 11 T 12 T 13 T 21 T 22 T 23 T 31 T 32 T 33 ] T=\begin{bmatrix} T_{xx}&T_{xy}&T_{xz} \\ T_{yx}&T_{yy}&T_{yz} \\ T_{zx}&T_{zy}&T_{zz}\end{bmatrix}=\begin{bmatrix} T_{11}&T_{12}&T_{13} \\ T_{21}&T_{22}&T_{23} \\ T_{31}&T_{32}&T_{33} \end{bmatrix} T=⎣⎡TxxTyxTzxTxyTyyTzyTxzTyzTzz⎦⎤=⎣⎡T11T21T31T12T22T32T13T23T33⎦⎤
单位矢量和单位阵：
e 1 = e x = ( 1 0 0 ) e_1=e_x=\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix} e1=ex=⎝⎛100⎠⎞
e 2 = e y = ( 0 1 0 ) e_2=e_y=\begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix} e2=ey=⎝⎛010⎠⎞
e 3 = e z = ( 0 0 1 ) e_3=e_z=\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix} e3=ez=⎝⎛001⎠⎞
I = [ 1 0 0 0 1 0 0 0 1 ] I=\begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{bmatrix} I=⎣⎡100010001⎦⎤
简单操作：
C b ⃗ = ( C b x C b y C b z ) C\vec b=\begin{pmatrix} C b_x \\ C b_y \\ C b_z\end{pmatrix} Cb=⎝⎛CbxCbyCbz⎠⎞
C T = [ C T x x C T x y C T x z C T y x C T y y C T y z C T z x C T z y C T z z ] C T=\begin{bmatrix} C T_{xx}&C T_{xy}&C T_{xz} \\ C T_{yx}&C T_{yy}&C T_{yz} \\C T_{zx}&C T_{zy}&C T_{zz}\end{bmatrix} CT=⎣⎡CTxxCTyxCTzxCTxyCTyyCTzyCTxzCTyzCTzz⎦⎤
内积
内积的基本原理：
a i ⃗ ⋅ b j ⃗ = a b i ⃗ ⋅ j ⃗ = 0 a i ⃗ ⋅ b k ⃗ = a b i ⃗ ⋅ k ⃗ = 0 a j ⃗ ⋅ b k ⃗ = a b j ⃗ ⋅ k ⃗ = 0 a i ⃗ ⋅ b i ⃗ = a b i ⃗ ⋅ i ⃗ = a b a j ⃗ ⋅ b j ⃗ = a b j ⃗ ⋅ j ⃗ = a b a k ⃗ ⋅ b k ⃗ = a b k ⃗ ⋅ k ⃗ = a b a i ⃗ ⋅ b i j ⃗ = a b i ⃗ ⋅ i j ⃗ = a b j ⃗ b i j ⃗ ⋅ a i ⃗ = a b i j ⃗ ⋅ i ⃗ = 0 a\vec i\cdot b\vec j=ab\vec i \cdot\vec j=0\\ a\vec i\cdot b\vec k=ab\vec i \cdot\vec k=0\\ a\vec j\cdot b\vec k=ab\vec j \cdot\vec k=0\\ a\vec i\cdot b\vec i=ab\vec i \cdot\vec i=ab\\ a\vec j\cdot b\vec j=ab\vec j \cdot\vec j=ab\\ a\vec k\cdot b\vec k=ab\vec k \cdot\vec k=ab\\a\vec i\cdot b\vec {ij}=ab\vec i\cdot \vec {ij}=ab\vec j\\b\vec {ij}\cdot a\vec {i}=ab\vec {ij}\cdot \vec {i}=0 ai⋅bj=abi⋅j=0ai⋅bk=abi⋅k=0aj⋅bk=abj⋅k=0ai⋅bi=abi⋅i=abaj⋅bj=abj⋅j=abak⋅bk=abk⋅k=abai⋅bij=abi⋅ij=abjbij⋅ai=abij⋅i=0
ϕ = a ⃗ ⋅ b ⃗ = ( a 1 i ⃗ + a 2 j ⃗ + a 3 k ⃗ ) ⋅ ( b 1 i ⃗ + b 2 j ⃗ + b 3 k ⃗ ) = a 1 b 1 + a 2 b 2 + a 3 b 3 = ∑ i = 1 3 a i b i = a i b i \phi=\vec a \cdot \vec b=(a_1\vec i+a_2\vec j+a_3\vec k)\cdot (b_1\vec i+b_2\vec j+b_3\vec k)=a_1b_1+a_2b_2+a_3b_3=\sum_{i=1}^3 a_ib_i=a_ib_i ϕ=a⋅b=(a1i+a2j+a3k)⋅(b1i+b2j+b3k)=a1b1+a2b2+a3b3=i=1∑3aibi=aibi
b ⃗ = a ⃗ ⋅ T = ( a 1 i ⃗ + a 2 j ⃗ + a 3 k ⃗ ) ⋅ ( T 11 i ⃗ i ⃗ + T 12 i ⃗ j ⃗ + T 13 i ⃗ k ⃗ + T 21 j ⃗ i ⃗ + T 22 j ⃗ j ⃗ + T 23 j ⃗ k ⃗ + T 31 k ⃗ i ⃗ + T 32 k ⃗ j ⃗ + T 33 k ⃗ k ⃗ ) = a 1 T 11 i ⃗ + a 1 T 12 j ⃗ + a 1 T 13 k ⃗ + a 2 T 21 i ⃗ + a 2 T 22 j ⃗ + a 2 T 23 k ⃗ + a 3 T 31 i ⃗ + a 3 T 32 j ⃗ + a 3 T 33 k ⃗ = ( a 1 T 11 + a 2 T 21 + a 3 T 31 a 1 T 12 + a 2 T 22 + a 3 T 32 a 1 T 13 + a 2 T 23 + a 3 T 33 ) = ∑ i = 1 3 ∑ j = 1 3 a i T i j e ⃗ j = a i T i j e ⃗ j \vec b=\vec a \cdot T=(a_1\vec i+a_2\vec j+a_3\vec k)\cdot (T_{11}\vec i\vec i+T_{12}\vec i\vec j+T_{13}\vec i\vec k+T_{21}\vec j\vec i+T_{22}\vec j\vec j+T_{23}\vec j\vec k+T_{31}\vec k\vec i+T_{32}\vec k\vec j+T_{33}\vec k\vec k)\\ =a_1T_{11}\vec i+a_1T_{12}\vec j+a_1T_{13}\vec k+a_2T_{21}\vec i+a_2T_{22}\vec j+a_2T_{23}\vec k+a_3T_{31}\vec i+a_3T_{32}\vec j+a_3T_{33}\vec k \\=\begin{pmatrix} a_1T_{11}+ a_2T_{21}+a_3T_{31}\\ a_1T_{12}+a_2T_{22}+a_3T_{32} \\ a_1T_{13}+a_2T_{23}+a_3T_{33}\end{pmatrix}=\sum_{i=1}^3\sum_{j=1}^3{a_iT_{ij}\vec e_j}=a_iT_{ij}\vec e_j b=a⋅T=(a1i+a2j+a3k)⋅(T11ii+T12ij+T13ik+T21ji+T22jj+T23jk+T31ki+T32kj+T33kk)=a1T11i+a1T12j+a1T13k+a2T21i+a2T22j+a2T23k+a3T31i+a3T32j+a3T33k=⎝⎛a1T11+a2T21+a3T31a1T12+a2T22+a3T32a1T13+a2T23+a3T33⎠⎞=i=1∑3j=1∑3aiTijej=aiTijej
b ⃗ = T ⋅ a ⃗ = ( T 11 i ⃗ i ⃗ + T 12 i ⃗ j ⃗ + T 13 i ⃗ k ⃗ + T 21 j ⃗ i ⃗ + T 22 j ⃗ j ⃗ + T 23 j ⃗ k ⃗ + T 31 k ⃗ i ⃗ + T 32 k ⃗ j ⃗ + T 33 k ⃗ k ⃗ ) ⋅ ( a 1 i ⃗ + a 2 j ⃗ + a 3 k ⃗ ) = a 1 T 11 i ⃗ + a 2 T 12 i ⃗ + a 3 T 13 i ⃗ + a 1 T 21 j ⃗ + a 2 T 22 j ⃗ + a 3 T 23 j ⃗ + a 1 T 31 j ⃗ + a 2 T 32 j ⃗ + a 3 T 33 j ⃗ = ( a 1 T 11 + a 2 T 12 + a 3 T 13 a 1 T 21 + a 2 T 22 + a 3 T 23 a 1 T 31 + a 2 T 32 + a 3 T 33 ) = ∑ i = 1 3 ∑ j = 1 3 a i T j i e ⃗ i = a i T j i e ⃗ i \vec b= T\cdot \vec a=(T_{11}\vec i\vec i+T_{12}\vec i\vec j+T_{13}\vec i\vec k+T_{21}\vec j\vec i+T_{22}\vec j\vec j+T_{23}\vec j\vec k+T_{31}\vec k\vec i+T_{32}\vec k\vec j+T_{33}\vec k\vec k)\cdot (a_1\vec i+a_2\vec j+a_3\vec k)\\=a_1T_{11}\vec i+a_2T_{12}\vec i+a_3T_{13}\vec i+a_1T_{21}\vec j+a_2T_{22}\vec j+a_3T_{23}\vec j+a_1T_{31}\vec j+a_2T_{32}\vec j+a_3T_{33}\vec j\\=\begin{pmatrix} a_1T_{11}+a_2T_{12}+a_3T_{13}\\ a_1T_{21}+a_2T_{22}+a_3T_{23} \\ a_1T_{31}+a_2T_{32}+a_3T_{33}\end{pmatrix}=\sum_{i=1}^3\sum_{j=1}^3{a_iT_{ji}\vec e_i}=a_iT_{ji}\vec e_i b=T⋅a=(T11ii+T12ij+T13ik+T21ji+T22jj+T23jk+T31ki+T32kj+T33kk)⋅(a1i+a2j+a3k)=a1T11i+a2T12i+a3T13i+a1T21j+a2T22j+a3T23j+a1T31j+a2T32j+a3T33j=⎝⎛a1T11+a2T12+a3T13a1T21+a2T22+a3T23a1T31+a2T32+a3T33⎠⎞=i=1∑3j=1∑3aiTjiei=aiTjiei
外积
方法取自：Tobias Holzmann《Mathematics, Numerics, Derivations and OpenFOAM》