Hirt 



The paper is divided into several sections. In the first section a simple 

 linear difference equation is used to demonstrate several important features of 

 finite-difference equations that we shall use in developing the MAC equations. 

 The following sections develop the basic MAC difference equations and subsidi- 

 ary details. The next-to-last section presents two applications involving free- 

 surface flows, while the final section makes several comments on the applica- 

 tion of high-speed computers to the study of turbulence. 



A LINEAR EQUATION 



We begin by investigating the properties of a simple finite-difference equa- 

 tion. Many results derived here are directly applicable to the Navier-Stokes 

 equations. 



Consider the differential equation 



a 2„ 



(1) 



which describes the convection and diffusion of a scalar function p(x,t). The 

 convection velocity u and positive diffusion coefficient v are assumed constant. 

 A simple, explicit, finite -difference approximation to Eq. (1) is 



where Pj" - p(jSx,nSt), Sx is the space increment, and St is the time incre- 

 ment. The simplicity of Eq. (2) does not guarantee that it is a good approxima- 

 tion to Eq, (1). Equation (2) may have solutions that exhibit computational insta- 

 bilities or other inaccuracies making it useless. 



Equation (1) is stable in the sense that its solutions are bounded and other- 

 wise well behaved. The stability properties of Eq. (2) can be determined by the 

 Fourier method proposed by von Neumann (2). Equation (2) has exponentially 

 growing solutions that oscillate in sign if (2ySt/Sx2) > i, and nonoscillating ex- 

 ponentially growing solutions M v < (u^St/2). In either case these growing so- 

 lutions in no way approximate the bounded solutions of Eq. (1). Thus, the in- 

 equalities 



—-r < 1 , V > ^u^St (3) 



ox'' ^ 



are stability conditions for the difference Eq. (2). For specified values of v, Sx, 

 and u these conditions define a range of St values that do not lead to exponen- 

 tially growing solutions. 



Stability conditions in Eq. (3) can be determined in another way. The alter- 

 native method we shall describe is more useful for our purposes than the linear 

 Fourier method, because it is also applicable to the nonlinear Navier-Stokes 



416 



