有限體積法

舉例

${\displaystyle \quad (1)\qquad \qquad {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.}$

${\displaystyle \rho =\rho \left(x,t\right)\ }$ 在這裡代表狀態變量， ${\displaystyle f=f\left(\rho \left(x,t\right)\right)\ }$ 代表的通量流量${\displaystyle \rho \ }$  。習慣上，${\displaystyle f\ }$ 正值代表向右流動，而${\displaystyle f\ }$ 負值代表向左流動。如果假設式（1）表示恆定面積的流動介質，則可以空间域 ${\displaystyle x\ }$  ，细分為數個網格單元以每個網格單元所佔的有限體積以${\displaystyle i\ }$ 作為標記 。對於特定的單元${\displaystyle i\ }$  ，我们可以定義該體積某物理量（ 壓力、溫度等 ）之通量流量平均值${\displaystyle {\rho }_{i}\left(t\right)=\rho \left(x,t\right)\ }$ 在時間${\displaystyle {t=t_{1}}\ }$ ${\displaystyle {x\in \left[x_{i-{\frac {1}{2}}},x_{i+{\frac {1}{2}}}\right]}\ }$  ，如式（2）

${\displaystyle \quad (2)\qquad \qquad {\bar {\rho }}_{i}\left(t_{1}\right)={\frac {1}{x_{i+{\frac {1}{2}}}-x_{i-{\frac {1}{2}}}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\rho \left(x,t_{1}\right)\,dx,}$

${\displaystyle \quad (3)\qquad \qquad {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{x_{i+{\frac {1}{2}}}-x_{i-{\frac {1}{2}}}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\rho \left(x,t_{2}\right)\,dx,}$

${\displaystyle \quad (4)\qquad \qquad \rho \left(x,t_{2}\right)=\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)\,dt,}$

${\displaystyle f_{x}={\frac {\partial f}{\partial x}}}$

${\displaystyle \quad (5)\qquad \qquad {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{\Delta x_{i}}}\int _{x_{i-{\frac {1}{2}}}}^{x_{i+{\frac {1}{2}}}}\left\{\rho \left(x,t_{1}\right)-\int _{t_{1}}^{t_{2}}f_{x}\left(x,t\right)dt\right\}dx.}$

${\displaystyle \quad (6)\qquad \qquad {\bar {\rho }}_{i}\left(t_{2}\right)={\bar {\rho }}_{i}\left(t_{1}\right)-{\frac {1}{\Delta x_{i}}}\left(\int _{t_{1}}^{t_{2}}f_{i+{\frac {1}{2}}}dt-\int _{t_{1}}^{t_{2}}f_{i-{\frac {1}{2}}}dt\right).}$

${\displaystyle f_{i\pm {\frac {1}{2}}}=f\left(x_{i\pm {\frac {1}{2}}},t\right)}$

${\displaystyle \quad (7)\qquad \qquad {\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{i+{\frac {1}{2}}}-f_{i-{\frac {1}{2}}}\right]=0,}$

一般守恆法則

${\displaystyle \quad (8)\qquad \qquad {{\partial {\mathbf {u} }} \over {\partial t}}+\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)={\mathbf {0} }.}$

${\displaystyle \quad (9)\qquad \qquad \int _{v_{i}}{{\partial {\mathbf {u} }} \over {\partial t}}\,dv+\int _{v_{i}}\nabla \cdot {\mathbf {f} }\left({\mathbf {u} }\right)\,dv={\mathbf {0} }.}$

${\displaystyle \quad (10)\qquad \qquad v_{i}{{d{\mathbf {\bar {u}} }_{i}} \over {dt}}+\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} },}$

${\displaystyle \quad (11)\qquad \qquad {{d{\mathbf {\bar {u}} }_{i}} \over {dt}}+{{1} \over {v_{i}}}\oint _{S_{i}}{\mathbf {f} }\left({\mathbf {u} }\right)\cdot {\mathbf {n} }\ dS={\mathbf {0} }.}$

相關文獻

• Eymard, R. Gallouët, T. R., (2000) The finite volume method Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
• Hirsch, C. (1990), Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley.
• Laney, Culbert B. (1998), Computational Gas Dynamics, Cambridge University Press.
• LeVeque, Randall (1990), Numerical Methods for Conservation Laws, ETH Lectures in Mathematics Series, Birkhauser-Verlag.
• LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
• Patankar, Suhas V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere.
• Tannehill, John C., et al., (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.
• Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
• Wesseling, Pieter (2001), Principles of Computational Fluid Dynamics, Springer-Verlag.

参考資料

1. ^ LeVeque, Randall. Finite Volume Methods for Hyperbolic Problems. 2002. ISBN 9780511791253.
2. ^ Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis. Applied Mathematical Modelling. 2000-06-01, 24 (7): 439–455. ISSN 0307-904X. doi:10.1016/S0307-904X(99)00047-5 （英语）.
3. ^ Ranganayakulu, C. (Chennu). Chapter 3, Section 3.1. Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2. OCLC 1006524487.