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and Quick Quickest This lecture is based upon the paper by B. P. Leonard, deriving the QUICK and QUICKEST methods. Note that we have already derived these methods via a different path in previous lectures. Take a look at Leonard s Equation 23. Renaming the variables to match my notation, we have: r = 1 ( C + R ) 1 ( L + R 2 C ) = 1 ( i + i +1 ) 1 ( i 1 + i +1 2 i ) 2 8 2 8 or 3 3 1 r = i +1 + i i 1 8 4 8 Look familiar? Notice that QUICK is one of those instances where you get something by thinking about practical bounds on the wave number in the stability analysis. Basic Approximations in Leonard s Derivation If you are having trouble with Equations 49 through 51, remember that they work from the assumption that the density is a quadratic function of x. For example, if I need to know the density near the face between volumes i and i+1, I work with the first three terms of a Taylor series expansion about the face. ( x ) = r + ( x x r ) x xr 2 1 2 + ( x xr ) 2 x2 xr To simplify notation Leonard introduced a new space variable . A positive sign of this new variable corresponds to the upstream direction. He s avoiding use of x in this context, since that denotes the length of the spatial mesh. The two derivatives in the expression are approximated by finite difference, giving: ( ) = r i +1 i x + 1 2 i +1 + i 1 2 i 1 = r GRADr + 2 CURVr 2 x 2 r = xr x Dealing with Time Derivatives Assume that a spatial density distribution is advected without change shape. in The time history of density at the right face can be obtained using the density profile at the beginning of the time step upstream of the face. The relationship between time and space variables in these two ways of viewing density is: d = V dt. So the time integral converts to a space integral as follows: t (t )dt = 0 V t ( )d 0 V t 12 r GRADr + CURVr d = 2 0 V t 2 1 3 r GRADr + CURVr = 2 23 0 (V t ) 2 CURV V t V t r GRADr + r 2 6 or in terms of the material Courant number c, we have t ( c x ) 2 CURV c x r (t )dt = c x r 2 GRADr + 6 0 Error Cancellation in the Presence of Source Terms Consider a mass conservation equation that might result from multi-phase or reacting flow: +V = S (x,t) t x If the mass source term S(x,t) is small compared to the mass flux term then the Quickest difference formula is a reasonable approximation. However, in many such problems a quasi-equilibrium exists with: V S (x,t) x Hence, the substitutions imbedded in Quickest: 2 = V , =V t x t2 2 2 3 , = V x2 t3 2 3 x3 are no longer valid. Use of a vanilla Quickest difference operator for flux terms does not give higher order accuracy. Be cautious of any numerical solution of flow equations that advertises both the Quickest difference method, and the ability to model multi-phase and/or reacting flows.

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