Reverse shooting

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

Example 9.5 Repeat Example 9.4 using reverse shooting. 

The next example illustrates the use of reverse shooting in solving a problem in nonisothermal axial dispersion and shows how Runge-Kutta integration can be applied to second-order ODEs. 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

The marching equation for reverse shooting. Equation (9.24), was developed using a first-order, backward difference approximation for dajdz, even though a second-order approximation was necessary for (faldz. Since the locations j—l, j, and j+ are involved an5rway, would it not be better to use a second-order, central difference approximation for dajdz ... 

The marching-ahead equation becomes a marching-backward equation. The method is called reverse shooting. The procedure is to guess aj = aout and then to set aj / = aj. The index j in Equation (9.30) begins at J — 2 and is decremented by 1 until / = 0 is reached. The reaction rate continues to be evaluated at the central, /th point. The test condition is whether ain is correct when calculated using the inlet boundary condition... 

EXAMPLE 9.4 Repeat Example 9.3 using reverse shooting. 

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