Numerical Solutions to Two-Point Boundary Value Problems

5 NUMERICAL SOLUTIONS TO TWO-POINT BOUNDARY VALUE PROBLEMS 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz.  

Apply finite difference approximations to Equation (9.15) using a backwards difference for da/d and a central difference for d a/d. The result is 

the value for the next, j+, point requires knowledge of two previous points,/ and j— 1. To calculate U2, we need to know both and uq. The boundary conditions. Equations (9.16) and (9.17), give neither of these directly. In finite difference form, the inlet boundary condition is 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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