欧拉方程：二维直角坐标，等温，微分形式，守恒形式

$$\begin{cases} \frac{\partial }{\partial t} \rho + \frac{\partial } {\partial x} (\rho u )+ \frac{\partial }{\partial y} (\rho v) = 0 \ \frac{\partial}{\partial t}(\rho u) + \frac{\partial }{\partial x} (\rho u^2 + p) + \frac{\partial }{\partial y} (\rho uv) = 0\ \frac{\partial}{\partial t}(\rho v) + \frac{\partial }{\partial x} (\rho uv) + \frac{\partial }{\partial y} (\rho v^2 + p) = 0 \end{cases}$$

$$p = p_0 + c^2(\rho - \rho_0)$$

欧拉方程：二维轴对称，等温，微分形式，非守恒形式

$$\begin{cases} \frac{\partial}{\partial t} \rho + \frac{\partial }{\partial r} (\rho u) + \frac{\partial}{\partial z} (\rho v) = -\frac{\rho u}{r}\ \frac{\partial}{\partial t} u + u \frac{\partial}{\partial r} u + v\frac{\partial}{\partial z} u = -\frac{1}{\rho}\frac{\partial}{\partial r}p\ \frac{\partial}{\partial t} v + u\frac{\partial}{\partial r} v + v\frac{\partial}{\partial z} v = -\frac{1}{\rho}\frac{\partial}{\partial z} p \end{cases}$$

欧拉方程：二维轴对称，等温，微分形式1，守恒形式

$$\begin{cases} \frac{\partial }{\partial t}\rho + \frac{\partial}{\partial r}(\rho u) + \frac{\partial}{\partial z}(\rho v) = - \frac{1}{r} (\rho u) \ \frac{\partial}{\partial t} (\rho u)+ \frac{\partial}{\partial r}(\rho u^2 + p) + \frac{\partial}{\partial z}(\rho uv) = -\frac{1}{r}(\rho u^2)\ \frac{\partial}{\partial t}(\rho v) + \frac{\partial}{\partial r}(\rho uv) + \frac{\partial}{\partial z}(\rho v^2 + p) = -\frac{1}{r} (\rho uv) \end{cases}$$

欧拉方程：二维轴对称，等温，微分形式2，守恒形式

$$\begin{cases} \frac{\partial}{\partial t} (r\rho) + \frac{\partial}{\partial r}(r \rho u) + \frac{\partial}{\partial z} (r\rho v)= 0\ \frac{\partial}{\partial t} (r \rho u) + \frac{\partial}{\partial r}(r \rho u^2 + r p) +\frac{\partial}{\partial z}(r\rho uv) = p\ \frac{\partial}{\partial t} (r \rho v) + \frac{\partial}{\partial r}(r \rho u v) +\frac{\partial}{\partial z}(r\rho v^2 + r p) = 0\ \end{cases}$$

欧拉方程：二维轴对称，等温，积分形式2，守恒形式

$$\dfrac{\partial U}{\partial t} + \dfrac{\partial F}{\partial r} + \dfrac{\partial G}{\partial z} = 0$$

$$U=\begin{pmatrix} r \rho\ r \rho u\ r \rho v \end{pmatrix}, F =\begin{pmatrix} r\rho u\ r\rho u^2 + r p\ r\rho u v \end{pmatrix}, G =\begin{pmatrix} r\rho v\ r\rho u v\ r\rho v^2 + rp \end{pmatrix}, S = \begin{pmatrix} 0\ p\ 0 \end{pmatrix}$$

$$A=\begin{pmatrix} 0 & 1 & 0\ -u^2 + a^2 & 2u & 0\ -uv & v & u \end{pmatrix}$$

$$L=\begin{pmatrix} 0 & 1 & 1\ 0 & u+a & u-a\ 1 & v & v \end{pmatrix}, R = \begin{pmatrix} \frac{2 a v}{-u + v - a} & \frac{u - v - a}{-u + v - a} & 1 \ \frac{v}{-u + v - a} & \frac{-1}{-u + v - a} & 0\ \frac{1}{-u + v - a} & \frac{-a- u}{-u + v - a} & 0 \end{pmatrix} $$

$$A^+ = \begin{pmatrix} 0 & 1 & 1\ 0 & u+a & u-a\ 1 & v & v \end{pmatrix} \begin{pmatrix} \lambda_1^+ & 0 & 0\ 0 & \lambda_2^+ & 0\ 0 & 0 & \lambda_3^+ \end{pmatrix}$$

波动方程：二维直角坐标，无耗散

$$\begin{cases} \frac{\partial}{\partial t} (\rho) + \frac{\partial}{\partial x} ( \rho_0 u) + \frac{\partial}{\partial y} (\rho_0 v) = 0\ \frac{\partial}{\partial t}(\rho_0 u) + \frac{\partial}{\partial x} (p) = 0 \ \frac{\partial}{\partial t}(\rho_0 v) + \frac{\partial}{\partial y} (p) = 0 \end{cases}$$

波动方程：二维轴对称，无耗散，守恒形式1

$$\begin{cases} \frac{\partial}{\partial t} \rho+ \frac{\partial}{\partial r}(\rho_0 u) + \frac{\partial }{\partial z}(\rho_0 v) = -\frac{1}{r} (\rho_0 u) \ \frac{\partial}{\partial t} (\rho_0 u) = - \frac{\partial} {\partial r} p \ \frac{\partial }{\partial t} (\rho_0 v) = - \frac{\partial}{\partial z} p \end{cases}$$

波动方程：二维轴对称，无耗散，守恒形式2

$$\begin{cases} \frac{\partial}{\partial t} (r\rho) + \frac{\partial}{\partial r}(r\rho_0 u) + \frac{\partial }{\partial z}(r\rho_0 v) = 0 \ \frac{\partial}{\partial t} (r\rho_0 u) + \frac{\partial} {\partial r} (r p) = p\ \frac{\partial }{\partial t} (r\rho_0 v) + \frac{\partial}{\partial z} (rp) = 0 \end{cases}$$

$$U=\begin{pmatrix} r\rho \ r\rho_0 u\ r\rho_0 v \end{pmatrix}, F = \begin{pmatrix} r\rho_0 u\ r [a^2(\rho - \rho_0) + p_0]\ 0\ \end{pmatrix}, G = \begin{pmatrix} r\rho_0 v\ 0\ rp \end{pmatrix}, S = \begin{pmatrix} 0\ p\ 0 \end{pmatrix}$$

注意常数项乘了r, 所以求导有值！

$$A = \dfrac{\partial F}{\partial U}= \begin{pmatrix} 0 & 1 & 0\ a^2 & 0 & 0\ 0 & 0 & 0 \end{pmatrix}, B = \dfrac{\partial G}{\partial U }= \begin{pmatrix} 0 & 0 & 1\ 0 & 0 & 0\ a^2 & 0 & 0 \end{pmatrix}$$

$$F= 0\begin{pmatrix}0\0\1\end{pmatrix} + \frac{1}{2a}(rp - ar\rho_0 u)\begin{pmatrix}-1\a\0\end{pmatrix} + \frac{1}{2a}(rp +ar\rho_0 u) \begin{pmatrix}1\a\0\end{pmatrix}$$$$x = y$$

$$A^+ = \begin{pmatrix} 0 & -\frac{1}{a} & \frac{1}{a}\ 0 & 1 & 1\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0\ 0 & 0 & 0\ 0 & 0 & a \end{pmatrix} \begin{pmatrix} 0 & 0 & 1\ -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a}{2} & \frac{1}{2} & 0\ \end{pmatrix}= \begin{pmatrix} \frac{a}{2} & \frac{1}{2} & 0\ \frac{a^2}{2} & \frac{a}{2} & 0 \ 0 & 0 & 0\ \end{pmatrix}$$

$$A^- = \begin{pmatrix} 0 & -\frac{1}{a} & \frac{1}{a}\ 0 & 1 & 1\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0\ 0 & -a & 0\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1\ -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a}{2} & \frac{1}{2} & 0\ \end{pmatrix}=\begin{pmatrix} -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a^2}{2} & -\frac{a}{2} & 0\ 0 & 0 & 0 \end{pmatrix}$$

$$F^+ = \begin{pmatrix} \frac{1}{2} r \rho a + \frac{1}{2} r \rho_0 u\ \frac{a}{2} r \rho a + \frac{a}{2} r \rho_0 u\ 0 \end{pmatrix}, F^- = \begin{pmatrix} -\frac{1}{2} r \rho a + \frac{1}{2} r \rho_0 u\ \frac{a}{2} r \rho a - \frac{a}{2} r \rho_0 u\ 0\end{pmatrix}$$

$$G^+ = \begin{pmatrix} \frac{1}{2} r \rho a + \frac{1}{2} r \rho_0 v\0\ \frac{a}{2} r \rho a + \frac{a}{2} r \rho_0 v\ \end{pmatrix}, G^- = \begin{pmatrix} -\frac{1}{2} r \rho a + \frac{1}{2} r \rho_0 v\0\ \frac{a}{2} r \rho a - \frac{a}{2} r \rho_0 v\ \end{pmatrix}$$

$$\dfrac{\partial U}{\partial t} + \dfrac{\partial F(U,r)}{\partial r} + \dfrac{\partial G(U,r)}{\partial z} = S$$

$$\dfrac{\partial U}{\partial t} + (\dfrac{\partial F}{\partial U}\dfrac{\partial U}{\partial r} + \dfrac{\partial F_0(r)}{\partial r}) + \dfrac{\partial G}{\partial z} = S$$

$$\dfrac{\partial U}{\partial t} + (A\dfrac{\partial U}{\partial r} + \dfrac{\partial F_0(r)}{\partial r}) + B\dfrac{\partial U}{\partial z} = S$$

$$\begin{split} U^{n+1}{i,j} = U^n{i,j} + S{i,j} dt - \frac{\Delta t}{\Delta r} A^+(U^n{i,j}) (U^n{i,j} - U^n{i-1,j}) - \frac{\Delta t}{\Delta r} A^-(U^n{i,j}) (U^n{i+1,j} - U^n{i,j}) \ - \frac{\Delta t}{\Delta z} B^+(U^n{i,j})(U^n{i,j}-U^n{i,j-1}) - \frac{\Delta t}{\Delta z} B^-(U^n{i,j})(U^n{i,j+1}-U^n_{i,j}) \end{split}$$

相互作用声波方程

$$\begin{cases}\frac{\partial} {\partial t} (\rho^\prime) + \bar{v}\cdot\nabla \rho^\prime + \rho_0 \nabla\cdot v^\prime = 0\ \frac{\partial}{\partial t}(\rho_0 v^\prime) + (\rho_0 v^\prime \cdot \nabla ) \bar{v} + (\rho_0 \bar{v} \cdot \nabla) v^\prime + (\rho_0 \bar{v}\cdot\nabla)\bar{v} = -\nabla p^\prime + \mu \nabla^2 v^\prime \end{cases}$$

$$\frac{\partial \rho^\prime}{\partial t} + \bar{u}\frac{\partial \rho^\prime}{\partial r} + \bar{v} \frac{\partial \rho^\prime}{\partial z} + \frac{\partial r\rho_0 u^\prime}{r\partial r} + \frac{\partial \rho_0v^\prime}{\partial z} = 0$$$$\frac{\partial}{\partial t} (\rho_0 u^\prime) + \rho_0u^\prime \frac{\partial r\bar{u}}{r\partial r} + \rho_0v^\prime\frac{\partial \bar{u}}{\partial z} + \rho_0 \bar{u} \frac{\partial r u^\prime}{r\partial r} + \rho_0\bar{v}\frac{\partial u^\prime}{\partial z} + \rho_0 \bar{u} \frac{\partial r\bar{u}}{r\partial r} + \rho_0\bar{v}\frac{\partial \bar{u}}{\partial z} = - \frac{\partial p}{\partial r}$$

$$\frac{\partial}{\partial t} (\rho_0 v^\prime) + \rho_0u^\prime \frac{\partial \bar{v}}{\partial r} + \rho_0v^\prime\frac{\partial \bar{v}}{\partial z} + \rho_0 \bar{u} \frac{\partial v^\prime}{\partial r} + \rho_0\bar{v}\frac{\partial v^\prime}{\partial z} + \rho_0 \bar{u} \frac{\partial \bar{v}}{\partial r} + \rho_0\bar{v}\frac{\partial \bar{v}}{\partial z} = - \frac{\partial p}{\partial z}$$

$$ \begin{cases} \dfrac{\partial }{\partial t}(r \rho^\prime) + \dfrac{\partial}{\partial r}( r \rho^\prime \bar{u} + r\rho_0 u^\prime) + \dfrac{\partial }{\partial z}(r\rho^\prime \bar{v}+r\rho_0v^\prime) = 0\ \frac{\partial}{\partial t} (r\rho_0 u^\prime) + \rho_0u^\prime \frac{\partial r\bar{u}}{\partial r} + r\rho_0v^\prime\frac{\partial \bar{u}}{\partial z} + \rho_0 \bar{u} \frac{\partial ru^\prime}{\partial r} + \rho_0r\bar{v}\frac{\partial u^\prime}{\partial z} + \rho_0 \bar{u} \frac{\partial r\bar{u}}{\partial r} + \rho_0r\bar{v}\frac{\partial \bar{u}}{\partial z} = - r\frac{\partial p}{\partial r}\ \dfrac{\partial}{\partial t} (r\rho_0 u^\prime) + \dfrac{\partial }{\partial r}(r \rho_0 \bar{u} u^\prime + r \rho_0 \bar{u}^2) + \dfrac{\partial }{\partial z}(r \rho_0 \bar{v} u^\prime + r\rho_0 \bar{v}\bar{u}) = -r\dfrac{\partial p}{\partial r} - \rho_0 \bar{u}u^\prime - \rho_0 \bar{u}^2 - \rho_0u^\prime \frac{\partial r\bar{u}}{\partial r}-r\rho_0v^\prime\frac{\partial \bar{u}}{\partial z} \end{cases}

$$

$$\begin{cases} \dfrac{\partial }{\partial t}(r \rho^\prime) + \dfrac{\partial}{\partial r}( r \rho^\prime \bar{u} + r\rho_0 u^\prime) + \dfrac{\partial }{\partial z}(r\rho^\prime \bar{v}+r\rho_0v^\prime) = 0 \ \dfrac{\partial}{\partial t} (r\rho_0 u^\prime) + \dfrac{\partial }{\partial r}(r \rho_0 \bar{u} u^\prime + r \rho_0 \bar{u}^2 + r p^\prime) + \dfrac{\partial }{\partial z}(r \rho_0 \bar{v} u^\prime + r\rho_0 \bar{v}\bar{u}) = p^\prime - 2\rho_0 \bar{u}u^\prime - \rho_0 \bar{u}^2 - r\rho_0u^\prime \frac{\partial \bar{u}}{\partial r}-r\rho_0v^\prime\frac{\partial \bar{u}}{\partial z}\ \dfrac{\partial}{\partial t} (r\rho_0 v^\prime) + \dfrac{\partial }{\partial r}(r \rho_0 \bar{u} v^\prime + r \rho_0 \bar{u}\bar{v}) + \dfrac{\partial }{\partial z}(r \rho_0 \bar{v} v^\prime + r\rho_0 \bar{v}^2 + r p^\prime) = -r\rho_0u^\prime \frac{\partial \bar{v}}{\partial r}-r\rho_0v^\prime\frac{\partial \bar{v}}{\partial z} \end{cases}$$

$$ U = \begin{pmatrix} r \rho^\prime\ r \rho_0 u^\prime\ r \rho_0 v^\prime \end{pmatrix} , F = \begin{pmatrix} r \rho_0 u^\prime + r\rho^\prime \bar{u}\ r\rho_0 u^\prime \bar{u} + r \rho_0 \bar{u}^2 + r p^\prime \ r \rho_0v^\prime\bar{u} + r\rho_0 \bar{u}\bar{v} \end{pmatrix}, G = \begin{pmatrix} r \rho_0 v^\prime + r \rho^\prime \bar{v}\ r \rho_0 u^\prime \bar{v} + r \rho_0 \bar{u}\bar{v}\ r \rho_0 v^\prime \bar{v} + r \rho_0 \bar{v}^2 + r p^\prime \end{pmatrix}, S = \begin{pmatrix} 0 \end{pmatrix}

$$

$$\bar{p} + p^\prime - p_0 = c^2 \rho^\prime\

$$rp^\prime = c^2 r\rho^\prime + r(p_0 - \bar{p})$$

$$ F = \begin{pmatrix} r \rho_0 u^\prime + r\rho^\prime \bar{u}\ r\rho_0 u^\prime \bar{u} + c^2 r \rho^\prime \ r \rho_0v^\prime\bar{u} + r\rho_0 \bar{u}\bar{v} \end{pmatrix}, F_0 = \begin{pmatrix} 0\ r(p_0 - \bar{p} - \rho \bar{u}^2 )\ r\rho_0 \bar{u}\bar{v} \end{pmatrix}$$$$$$\begin{cases}\frac{\partial} {\partial t} (\rho^\prime) +\nabla \cdot (\rho^\prime\bar{v} + \rho_0 v^\prime) = 0\ \frac{\partial}{\partial t}(\rho_0 v^\prime) + \nabla\cdot({c^2\rho^\prime}\mathbf{I} + \rho_0 v^\prime\bar{v}) = -(\rho_0 v^\prime \cdot \nabla ) \bar{v} +\nabla\cdot (\bar{p}\mathbf{I} - \rho_0 \bar{v}\bar{v})+ \mu \nabla^2 v^\prime \end{cases}$$

$$$$A= \begin{pmatrix} \bar{u} & 1 & 0 \ c^2 & \bar{u} & 0 \ 0 & 0 & \bar{u} \end{pmatrix}$$$$

$$$$\begin{cases}\frac{\partial} {\partial t} (\rho^\prime) + \bar{v}\cdot\nabla \rho^\prime + \rho_0 \nabla\cdot v^\prime = 0\ \frac{\partial}{\partial t}(\rho_0 v^\prime) + (\rho_0 v^\prime \cdot \nabla ) \bar{v} + (\rho_0 \bar{v} \cdot \nabla) v^\prime + (\rho_0 \bar{v}\cdot\nabla)\bar{v} = -\nabla (c^2\rho^\prime + p_0 -\bar{p})+ \mu \nabla^2 v^\prime \end{cases}$$$$

$$$$\nabla\cdot (\bar{p}\mathbf{I} - \rho_0 \bar{v}\bar{v})$$ $$ 互相抵消$$

$$\begin{cases} \dfrac{\partial }{\partial t}(r \rho^\prime) + \dfrac{\partial}{\partial r}( r \rho^\prime \bar{u} + r\rho_0 u^\prime) + \dfrac{\partial }{\partial z}(r\rho^\prime \bar{v}+r\rho_0v^\prime) = 0 \ \dfrac{\partial}{\partial t} (r\rho_0 u^\prime) + \dfrac{\partial }{\partial r}(r \rho_0 \bar{u} u^\prime + r c^2 \rho^\prime) + \dfrac{\partial }{\partial z}(r \rho_0 \bar{v} u^\prime) = r\dfrac{\partial \bar{p}}{\partial r}+(c^2\rho^\prime + \rho_0 \bar{u}u^\prime) - ( \rho_0u^\prime \dfrac{\partial r\bar{u}}{\partial r} +\rho_0v^\prime\dfrac{\partial r\bar{u}}{\partial z}) - (\rho_0\bar{u}\dfrac{\partial r \bar{u}}{\partial r} + \rho_0\bar{v}\dfrac{\partial r\bar{u}}{\partial z})\ \dfrac{\partial}{\partial t} (r\rho_0 v^\prime) + \dfrac{\partial }{\partial r}(r \rho_0 \bar{u} v^\prime ) + \dfrac{\partial }{\partial z}(r \rho_0 \bar{v} v^\prime + r c^2\rho^\prime) = r\dfrac{\partial \bar{p}}{\partial z}-(\rho_0u^\prime \dfrac{\partial r\bar{v}}{\partial r}+\rho_0v^\prime\dfrac{\partial r\bar{v}}{\partial z}) - (\rho_0\bar{u}\dfrac{\partial r \bar{v}}{\partial r} + \rho_0\bar{v}\dfrac{\partial r\bar{v}}{\partial z})\end{cases}$$

用矢通量分裂格式

$$ \dfrac{\partial U} {\partial t} + A \dfrac{\partial U}{\partial r} + B \dfrac{\partial U} {\partial z} = S

$$

$$$$A= \begin{pmatrix} \bar{u} & 1 & 0 \ c^2 & \bar{u} & 0 \ 0 & 0 & \bar{u} \end{pmatrix}, B = \begin{pmatrix} \bar{v} & 0 & 1\ 0 & \bar{v} & 0\ c^2 & 0 & \bar{v} \end{pmatrix}$$$$

特征值u, -c+u, c+u

$$A^+ = \begin{pmatrix} 0 & -\frac{1}{a} & \frac{1}{a}\ 0 & 1 & 1\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{u+|u|}{2} & 0 & 0\ 0 & 0 & 0\ 0 & 0 & c+u \end{pmatrix} \begin{pmatrix} 0 & 0 & 1\ -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a}{2} & \frac{1}{2} & 0\ \end{pmatrix}= \begin{pmatrix} \frac{c+u}{2} & \frac{c+u}{2c} & 0\ \frac{c(c+u)}{2} & \frac{c+u}{2} & 0 \ 0 & 0 & \frac{u+|u|}{2}\ \end{pmatrix}$$

$$A^- = \begin{pmatrix} 0 & -\frac{1}{a} & \frac{1}{a}\ 0 & 1 & 1\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{u-|u|}{2} & 0 & 0\ 0 & u-c & 0\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 1\ -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a}{2} & \frac{1}{2} & 0\ \end{pmatrix}= \begin{pmatrix} \frac{-c+u}{2} & \frac{c-u}{2c} & 0\ \frac{c(c-u)}{2} & \frac{-c+u}{2} & 0 \ 0 & 0 & \frac{u-|u|}{2}\ \end{pmatrix}$$

然后$$

$$F^{+} =\begin{pmatrix} r\rho^\prime (\frac{c+\bar{u}}{2}) + r\rho_0 u^\prime (\frac{c+\bar{u}}{2c})\ r\rho^\prime\frac{c(c+\bar{u})}{2} + r\rho_0 u^\prime(\frac{c+\bar{u}}{2})\ r\rho_0 v^\prime (\frac{\bar{u}+|\bar{u}|}{2})\ \end{pmatrix}, F ^{-}=\begin{pmatrix} r\rho^\prime (\frac{-c+\bar{u}}{2}) + r\rho_0u^\prime(\frac{c+\bar{u}}{2c})\ r\rho^\prime (\frac{c(c-\bar{u})}{2})+r\rho_0u^\prime(\frac{-c+\bar{u}}{2})\ r\rho_0 v^\prime(\frac{\bar{u}-|\bar{u}|}{2})

\end{pmatrix}$$

$$U^{n+1}{i,j} = U^n{i,j} + \Delta t S^n{i,j}+\frac{\Delta t}{\Delta r}(F^{n^+}{i,j}-F^{n^+}{i-1,j}) + \frac{\Delta t}{\Delta r}(F^{n^-}{i+1,j}-F^{n^-}{i,j}) +\frac{\Delta t}{\Delta z}(G^{n^+}{i,j}-G^{n^+}{i,j-1}) + \frac{\Delta t}{\Delta z}(G^{n^-}{i,j+1}-G^{n^-}_{i,j})$$

Mac-Cormack 格式，

预测步

$$U{i,j}^{n+1/2} = U{i,j}^n + S^n{i,j} \Delta t - \dfrac{\Delta t}{\Delta r} [F(U^{n}{i+1,j}) - F(U^n{i,j})] - \dfrac{\Delta t}{\Delta z} [G(U^{n}{i,j+1}) - G(U^n_{i,j})]$$

矫正步

$$U^{n+1}{i,j} = \frac{1}{2}[U^n{i,j} + U^{n+1/2}{i,j}] + \frac{1}{2} S^{n+1/2}{i,j} \Delta t - \dfrac{\Delta t}{\Delta r} [F(U^{n+1/2}{i,j}) - F(U^{n+1/2}{i-1,j})]- \dfrac{\Delta t}{\Delta z} [G(U^{n+1/2}{i,j}) - G(U^{n+1/2}{i,j-1})]$$

声波的反射边界条件：

玻璃声阻抗：1.18g/cm^3*2730m/s

水的声阻抗：998.2km/m3*1482m/s

空气：1.20kg/m3*343.21m/s

声源：

$$\begin{cases} \end{cases}$$

对称轴：z-coordinate

$$\begin{cases} \frac{\partial u}{\partial r}=\frac{\partial v}{\partial r}=\frac{\partial p}{\partial r} = 0 \ v_r = 0 \end{cases}$$

墙：

$$\begin{cases} \mathbf{v}\cdot \mathbf{n} = 0 \end{cases}$$

水到空气

$$\begin{cases} v = 0 \end{cases}$$

无相互作用的声波方程(柱坐标)

$$\begin{cases} \frac{\partial }{\partial t}\rho^\prime + \frac{\partial}{r\partial r}(r \rho_0 u^\prime) + \frac{\partial}{\partial z}(\rho_0 v^\prime) = 0\ \frac{\partial}{\partial t}(\rho_0 u) = -\frac{\partial}{\partial r} p^\prime\ \frac{\partial}{\partial t} (\rho_0 v) = -\frac{\partial}{\partial z} p^\prime \end{cases}$$

$$\begin{cases} \dfrac{\partial }{\partial t}(r\rho^\prime) + \dfrac{\partial}{\partial r} (r\rho_0 u^\prime) + \dfrac{\partial}{\partial z}(r\rho_0 v^\prime) = 0\ \dfrac{\partial }{\partial t}(r\rho_0 u^\prime) + \dfrac{\partial}{\partial r}(rp^\prime) = p^\prime\ \dfrac{\partial}{\partial r} (r\rho_0 v^\prime) + \dfrac{\partial}{\partial z}(rp^\prime) = 0 \end{cases}$$

吸收边界条件(一般形式)

$$\frac{\partial} {\partial x} \phi - \frac{1}{c}\frac{\partial}{\partial t} \phi = 0$$

$$\Phi^{n+1}{N+1/2} - \Phi^{n}{N+1/2} + \frac{c\Delta t}{\Delta x} (\Phi^n{N+1} - \Phi^n{N}) = 0 $$

$$\Phi^{n+1}{N+1} = \Phi^{n+1}{N} + \frac{c\Delta t/\Delta x - 1}{c\Delta t/\Delta x + 1} (\Phi^{n+1}{N+1} - \Phi^n{N}) $$

吸收边界条件

$$\dfrac{\partial U} {\partial t} + A \dfrac{\partial U}{\partial r} + B\dfrac{\partial U} {\partial z} = 0$$

$$\begin{cases} \dfrac{\partial }{\partial t}(r\rho^\prime) + \dfrac{\partial}{\partial r} (r\rho_0 u^\prime) + \dfrac{\partial}{\partial z}(r\rho_0 v^\prime) = 0\ \dfrac{\partial }{\partial t}(r\rho_0 u^\prime) + \dfrac{\partial}{\partial r}(rp^\prime) = p^\prime\ \dfrac{\partial}{\partial r} (r\rho_0 v^\prime) + \dfrac{\partial}{\partial z}(rp^\prime) = 0 \end{cases}$$

其中 $$p^\prime = a^2 \rho^\prime$$，进一步得到

$$L_A\dfrac{\partial U}{\partial t} + L_AB\dfrac{\partial U}{\partial z} + \text{Diag}(A) L_A\dfrac{\partial U}{\partial r} = 0$$

$$U = \begin{pmatrix} r\rho\ r\rho_0 u\ r\rho_0 v\ \end{pmatrix}, L_A = \begin{pmatrix} 0 & 0 & 1\ -\frac{a}{2} & \frac{1}{2} & 0\ \frac{a}{2} & \frac{1}{2} & 0\ \end{pmatrix}, L_AB = \begin{pmatrix} a^2 & 0 & 0\ 0 & 0 & -\frac{a}{2} \ 0 & 0 & \frac{a}{2} \end{pmatrix},L_A A = \begin{pmatrix} 0 & 0 & 0\ \frac{a^2}{2} & -\frac{a}{2} & 0\ \frac{a^2}{2} & \frac{a}{2} & 0\ \end{pmatrix}$$

$$\begin{cases} \frac{\partial}{\partial t}(r\rho_0 v)+a^2 \frac{\partial }{\partial z} (r\rho)= 0\ \frac{\partial}{\partial t}(\frac{a}{2}r\rho -\frac{1}{2}r\rho_0u) + \frac{\partial}{\partial z} (\frac{a}{2}r\rho_0 v) -\frac{\partial}{\partial r} (\frac{a^2}{2}r\rho - \frac{a}{2}r\rho_0 u) = 0\ \frac{\partial}{\partial t}(\frac{a}{2}r\rho + \frac{1}{2}r\rho_0 u) + \frac{\partial}{\partial z}(\frac{a}{2} r\rho_0 v) + \frac{\partial }{\partial r}(\frac{a^2}{2}r\rho + \frac{a}{2}r\rho_0 u ) = 0
\end{cases}$$

记$$L_1 = \frac{\partial}{\partial r}(a r \rho - r \rho_0 u), L_2 = \frac{\partial}{\partial r} (ar\rho + r\rho_0 u) $$

吸收边界条件，令$$L_1 = 0$$

$$\begin{cases} \rho0 \dfrac{r{N+1/2} ( v^{n+1}{N+1/2} -v^{n}{N+1/2})}{\Delta t} + a^2 \dfrac{r{N+1/2}(\rho^n{N+1}-\rho^n{N})}{\Delta z} = 0\ a\dfrac{r{N+1} \rho^n{N+1} - r{N} \rho^nN}{\Delta r} = \dfrac{ r{N+1/2}( \rho^{n+1}{N+1/2} -\rho^{n}{N+1/2} ) }{\Delta t} \a\dfrac{r{N+1} u^n{N+1} - r{N} u^n_N}{\Delta r} = \dfrac{ r{N+1/2}( u^{n+1}{N+1/2} -u^{n}{N+1/2}) }{\Delta t} \ \end{cases}$$