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

微分形式

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

积分形式

$$\dfrac{d}{dt}\int U(t; r, z) d\Omega + \int_{\partial \Omega}(F(U) dz + G(U) dr) = \int S d\Omega$$

有限体积法

对积分形式的方程进行差分离散

$$\dfrac{\bar{U}(t{n+1}, r_i, z_j)- \bar{U}(t{n}, ri, z_j)} {\Delta t} + \dfrac{F(t_n, r{i+1/2}, zj) - F(t_n, r{i-1/2}, zj) }{\Delta r} + \dfrac{G(t_n, r{i}, z{j+1/2}) - G(t_n, r_i, z{j-1/2}) }{\Delta z} = \bar{S}(ri, z_j)$$$$\bar{U}(t{n+1}, ri, z_j) = \bar{U}(t{n}, ri, z_j) + \bar{S}(r_i, z_j) {\Delta t} - \dfrac{\Delta t}{\Delta r}[F(t_n, r{i+1/2}, zj) - F(t_n, r{i-1/2}, zj)] - \dfrac{\Delta t}{\Delta z}[G(t_n, r{i}, z{j+1/2}) - G(t_n, r_i, z{j-1/2})]$$

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

迎风格式

1) 当 a > 0

$$\begin{cases} F^n{i-1/2,j} = F(U(t_n, r_i, z_j))\ F^n{i+1/2,j} = F(U(tn, r{i+1},z_j)) \end{cases}$$

2) 当 a < 0

$$\begin{cases} F^n{i-1/2,j} = F(U(t_n, r_i, z_j))\ F^n{i+1/2,j} = F(U(tn, r{i+1},z_j)) \end{cases}$$

Godunov-Riemann 求解器

$$\begin{cases} F^n{i-1/2,j} = F(U(t_n, r{i-1/2}, zj))\ F^n{i+1/2,j} = F(U(tn, r{i+1/2},z_j)) \end{cases}$$

线性近似

状态方程（等熵 ）

$$\begin{cases} (\rho/\rho_0)^n = \frac{K_0 + n (p - p_0)}{K_0}\ K_0 = {\rho} \Big(\frac{\partial p}{\partial \rho}\Big)_T\ \frac{\partial p}{\partial \rho} = \frac{K_0}{\rho_0}(\frac{\rho}{\rho_0})^{n-1} \end{cases}$$

Tait equation (https://en.wikibooks.org/wiki/Engineering_Acoustics/The_Acoustic_Parameter_of_Nonlinearity#cite_note-Beyer_2008-3\)

$$\begin{cases} V = V_0 - 0.315 V_0 \log \frac{B + P}{B + P_0} \ B = 2668.0 + 19.876 t - 0.311 t^2 + 1.778\text{e-3} t^3 \end{cases}$$

比较：气体状态方程

$$\begin{cases} p \rho^{-\gamma} = C\ a^2 := \Big(\dfrac{\partial p}{\partial \rho}\Big)_s = \gamma \dfrac{ p_0}{\rho_0} \end{cases}$$

所以，

$$F = \begin{pmatrix} r \rho u \ r \rho u^2 + rp \ r \rho uv
\end{pmatrix} = \begin{pmatrix} U_2 \ (U_2^2/U_1) + r p\ U_2U_3/U_1 \end{pmatrix}, G = \begin{pmatrix} r \rho v \ r \rho uv \ r \rho v^2 + r p \end{pmatrix} = \begin{pmatrix} U_3 \ U_2U_3/U_1 \ U_3^2/U_1 + r p \end{pmatrix} $$

雅可比矩阵 $$A= \frac{\partial F}{\partial U}, B= \frac{\partial G}{\partial U}$$

$$A = \begin{pmatrix} 0 & 1& 0 \ a^2(\frac{\rho} {\rho_0})^{n-1} - u^2& 2u& 0\ -uv& v& u\ \end{pmatrix}, B = \begin{pmatrix} 0& 0& 1\ -uv& v& u\ a^2(\frac{\rho} {\rho_0})^{n-1} - v^2 & 0& 2v\ \end{pmatrix}$$

有特征值

$$\lambda{A1} = u - \sqrt{a^2 (\frac{\rho}{\rho_0})^{n-1}}, \lambda{A2} = u,\lambda_{A3} = u + \sqrt{a^2 (\frac{\rho}{\rho_0})^{n-1}}$$

$$\lambda{B1} = v - \sqrt{a^2 (\frac{\rho}{\rho_0})^{n-1}}, \lambda{B2} = v,\lambda_{B3} = v + \sqrt{a^2 (\frac{\rho}{\rho_0})^{n-1}}$$

线性近似微分方程

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

Roe 格式

$$\begin{split} \bar{U}^{n+1}{i,j} = \bar{U}^{n}{i,j} + \bar{S}^{n}{i,j} \Delta t - \dfrac{\Delta t}{\Delta r} A(U{ij},U{i-1,j}) (U{i,j} - U{i-1,j}) - \dfrac{\Delta t}{\Delta r} A(U{i+1,j},U{i,j}) (U{i+1,j}- U{i,j}) \ - \dfrac{\Delta t}{\Delta z} B(U{i,j},U{i,j-1})(U{i,j} - U{i,j-1})- \dfrac{\Delta t}{\Delta z} B(U{i,j+1},U{i,j})(U{i,j+1} - U_{i,j})\end{split}$$

求解器