Numerical Simulation of Viscous Incompressible Fluid Flows 



equations. The method is based on an examination of truncation errors. Each 

 term in Eq. (2) is expanded in a Taylor series about the point x = jSx, t = nSt, 

 e.g., 



■" .... ox ^ Bx^ ^ , • ' 



Collecting terms, Eq. (2) becomes 



-L. + u — - V = - -^ + 0(Sx^bt2) , (4) 



3t dx 3x2 - Bt2 



where all second and higher order terms in &x and &t are represented by the 

 order symbol 0(ox2, St 2). The zero-order terms on the left-hand side of Eq, 

 (4) are the original differential Eq. (1). All terms on the right-hand side of Eq. 

 (4) are called truncation errors. These terms are responsible for the differ- 

 ence between solutions of the difference Eq. (2) and solutions of the differential 

 Eq. (1). This observation is important, because it suggests that the stability 

 conditions in Eq. (3) might be obtained directly from the truncation errors. 

 That this is indeed possible, at least approximately, has been shown in Ref. 3. 

 The prescription developed there is to keep only the lowest order even and odd 

 derivative terms with respect to each independent variable. As applied to Eq. 

 (4), this means keeping the first and second derivative terms with respect to 

 both x and t. After a slight rearrangement of terms, we have 



— V + •— +ur^=0, . (5) 



2 Bt2 Bx2 Bt Bx 



which is not identical to Eq, (1), the equation we set out to approximate. Equa- 

 tion (5) is a hyperbolic equation with characteristic lines whose slopes are 



(dx/dt) = ±(2v/6t)*/2. 



Similarly, difference Eq. (2) propagates information into a region of the x-t 

 plane bounded by lines whose slopes are (dx/dt = +Sx/St) . If difference Eq. (2) 

 is to have a solution approximating the solution of its counterpart in Eq. (5), 

 then its "region of influence" must at least include the region of influence of 

 Eq. (5), i.e., 



'i^y >2. . - . "• ' ^' ' (6) 



St/ St 



Courant et al. (4) have shown that a violation of this type of region-of-influence 

 condition leads to oscillating and exponentially growing solutions for the differ- 

 ence equation. This is also true in our case, since Eq. (6) is exactly the first 

 stability condition in Eq. (3). 



The condition in Eq. (6) can be given a physical interpretation. It states 

 that a wave disturbance must not travel more than one space increment (Sx in 

 one time step ht. We would expect this condition to be necessary for accuracy, 

 since difference Eq. (2) relates space location j only to neighboring locations 



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