計算手順と補正係数の決め方

2次元の FCT は1次元のものをそのまま拡張することで得ることができる. (18)を2次元形に拡張すると,

$$\rho _{i.j}^{n+1} = \rho _{i.j}^{n} - \frac{1}{\Delta x\Delta y}\{[C_{i+\frac{1}{2},j}F_{i+\frac{1}{2},j}^{H} + (1-C_{i+\frac{1}{2},j})F_{i+\frac{1}{2},j}^{L}]$$

$$- [C_{i-\frac{1}{2},j}F_{i-\frac{1}{2},j}^{H}+ (1-C_{i-\frac{1}{2},j})F_{i-\frac{1}{2},j}^{L}],$$

$$+ [C_{i,j+\frac{1}{2}}G_{i,j+\frac{1}{2}}^{H}+ (1-C_{i,j+\frac{1}{2}})G_{i,j+\frac{1}{2}}^{L}]$$

$$- [C_{i,j-\frac{1}{2}}G_{i,j-\frac{1}{2}}^{H}+ (1-C_{i,j-\frac{1}{2}})G_{i,j-\frac{1}{2}}^{L}]\},$$ (33')

となる. 補正係数 $C_{i\pm \frac{1}{2},j}, C_{i,j\pm \frac{1}{2}}$ を決める手順は(19)$\sim$(32)と同様である.

1. antidifuusive フラックスに対しあらかじめ以下の条件を課しておく.
   

   $$A_{i+\frac{1}{2},j} =0 \mbox{if} A_{i+\frac{1}{2},j}(\rho _ {i+1,j}^{td} - \rho _{i,j}^{td}) < 0,$$

   $$\mbox{and either} A_{i+\frac{1}{2},j}(\rho _ {i+2,j}^{td} - \rho _{i+1,j}^{td}) < 0,$$

   $$\mbox{or} A_{i+\frac{1}{2},j}(\rho _ {i,j}^{td} - \rho _{i-1,j}^{td}) < 0,$$

   $$A_{i,j+\frac{1}{2}} =0 \mbox{if} A_{i,j+\frac{1}{2}}(\rho _ {i,j+1}^{td} - \rho _{i,j}^{td}) < 0,$$

   $$\mbox{and either} A_{i,j+\frac{1}{2}}(\rho _ {i,j+2}^{td} - \rho _{i,j+1}^{td}) < 0,$$

   $$\mbox{or} A_{i,j+\frac{1}{2}}(\rho _ {i,j}^{td} - \rho _{i,j-1}^{td}) < 0.$$ (19')

   
   

2. $\rho _{i}^{max}, \rho _{i}^{min}$ を以下のようにして与える.
   

   $$\rho _{i,j}^{max} = \mbox{max}(\rho _{i-1,j}^{td}, \; \rho _ {i,j}^{td}, \; \rho _{i+1,j}^{td}, \; \rho _{i,j-1}^{td}, \; \rho _{i,j+1}^{td}),$$ (20')

   $$\rho _{i,j}^{min} = \mbox{min}(\rho _{i-1,j}^{td}, \; \rho _ {i,j}^{td}, \; \rho _{i+1,j}^{td}, \; \rho _{i,j-1}^{td}, \; \rho _{i,j+1}^{td}).$$ (21')

   
   

   または clipping を抑えるために,
   

   $$\rho _{i,j}^{a} = \mbox{max}(\rho _{i,j}^{n}, \; \rho _ {i,j}^{td}),$$ (22')

   $$\rho _{i}^{max} = \mbox{max}(\rho _{i-1,j}^{a}, \; \rho _ {i,j}^{a} , \; \rho _{i+1,j}^{a}, \; \rho _{i,j-1}^{a}, \; \rho _{i,j+1}^{a}),$$ (23')

   $$\rho _{i,j}^{b} = \mbox{min}(\rho _{i,j}^{n}, \; \rho _ {i,j}^{td}),$$ (24')

   $$\rho _{i,j}^{min} = \mbox{min}(\rho _{i-1,j}^{b}, \; \rho _ {i,j}^{b}, \; \rho _{i+1,j}^{b}, \; \rho _{i,j-1}^{b}, \; \rho _{i,j-1}^{b}),$$ (25')

   
   

   とする.

3. 以下の3つの量を用意する.
   

   $$P_{i,j}^{+} = \mbox{格子点 $i,j$\ に入る全ての antidiffusive フ ラックスの和}$$

   $$= \mbox{max}(0, A_{i-\frac{1}{2},j}) - \mbox{min}(0, A_{i+\frac{1}{2},j})$$

   $$+ \mbox{max}(0, A_{i,j-\frac{1}{2}}) - \mbox{min}(0, A_{i,j+\frac{1}{2}}),$$ (26')

   $$Q_{i,j}^{+} = (\rho _{i,j}^{max}-\rho _{i,j})\Delta x \Delta y,$$ (27')

   $$R_{i,j}^{+} = \left\{ \begin{array}{lcl} \mbox{min}(1, Q_{i.j}^{+}/P_{i,j}^{+})... ...mbox{if} & P_{i,j}^{+}>0, \\ 0 & \mbox{if} & P_{i,j}^{+}=0. \end{array}\right.$$ (28')

   
   

4. 同様に以下の3つの量を用意する.
   

   $$P_{i,j}^{-} = \mbox{格子点 $i,j$\ から出て行く全ての antidiffusive フラックスの和}$$

   $$= \mbox{max}(0, A_{i+\frac{1}{2},j}) - \mbox{min}(0, A_{i-\frac{1}{2},j})$$

   $$+ \mbox{max}(0, A_{i,j+\frac{1}{2}}) - \mbox{min}(0,A_{i,j-\frac{1}{2}}),$$ (29')

   $$Q_{i,j}^{-} = (\rho _{i,j}-\rho _{i,j}^{min})\Delta x \Delta y ,$$ (30')

   $$R_{i,j}^{-} = \left\{ \begin{array}{lcl} \mbox{min}(1, Q_{i,j}^{-}/P_{i,j}^{-})... ...mbox{if} & P_{i,j}^{-}>0, \\ 0 & \mbox{if} & P_{i,j}^{-}=0. \end{array}\right.$$ (31')

   
   

5. 補正係数を以下のように与える.
   

   $$C_{i+\frac{1}{2},j} = \left\{ \begin{array}{lcl} \mbox{min}(R_{i+1,j}^{+}, R_{i,j}^{-})... ...}^{+}, R_{i+1,j}^{-}) & \mbox{if} & A_{i+\frac{1}{2},j} < 0. \end{array}\right.$$ (32')

   $$C_{i,j+\frac{1}{2}} = \left\{ \begin{array}{lcl} \mbox{min}(R_{i,j+1}^{+}, R_{i,j}^{-})... ...}^{+}, R_{i,j+1}^{-}) & \mbox{if} & A_{i,j+\frac{1}{2}} < 0. \end{array}\right.$$