Hirt 



j ± 1. On the other hand, implicit finite-difference equations can avoid this type 

 of instability, since they require all space locations to be dependent on one an- 

 other. The region-of -influence condition will be called the wave propagation 

 condition for stability. 



The second stability condition in Eq. (3) is also obtainable from Eq. (4) if 

 the term proportional to St is expressed in terms of space derivatives. Using 

 Eq. (4), we have 



92p 3 2 3 2 34 



— -= u2 — -- 2^u — - + ^2 ^:+ 0(St) . (7) 



3t2 3x2 3x3 3x'' 



Inserting Eq. (7) into Eq. (4) yields 



-!—\ u -^- [v - iu2St + St zvu ^ Stz/2 + 0(St2, Sx2) . (8) 



3t 3x \ ^ / 3x2 3x3 2 3^4 



Keeping the lowest order even and odd derivative terms with respect to each in- 

 dependent variable gives us 



^.„|l=f.-iu^8t)^. (9) 



3t 3x \ ^ ] 3x2 



This truncated equation is similar to Eq. (1), except that it has a different diffu- 

 sion coefficient. If St is too large, the diffusion coefficient in Eq. (9) is negative, 

 and Eq. (9) then has exponentially growing solutions. For bounded solutions, the 

 diffusion coefficient must remain positive: 



V > i-u2St . (10) 



The condition in Eq. (10) is the second stability condition in Eq. (3). It states 

 that some v diffusion is necessary to keep the difference equation stable. 



Equation (9) also has a bearing on the accuracy of Eq. (2). For a given 

 value of V the effective diffusion coefficient is, to the terms of order 5x2 and 

 St^, V - (u2st/2), which increases as St decreases. Solutions of Eq. (2) are 

 smoother as St decreases, but for finite values of St , solutions of Eq. (2) are 

 subject to less diffusion than solutions of Eq. (1). 



All observations made about the linear difference equation can be applied to 

 difference equations in general. We shall make use of the wave propagation and 

 positive diffusion coefficient conditions to establish stability conditions for the 

 MAC method. Some important comments also will be made about diffusion-like 

 truncation errors in the MAC equations. 



418 



