Convergence criteria, absolute

Care needs to be taken if some components are present in trace quantities. If an estimated concentration is 0.5 ppm and the calculated value is 1 ppm, the scaled error is 100%. This is much too large an error for most variables and yet the absolute error might be acceptable for a trace component. In other situations, it might be necessary to define trace components with a high precision. A trace component threshold can be set, below which the convergence criterion is ignored. 

The iterative procedure is considered to give a converged solution if the absolute normalized residuals for all the variables as well as the mass source b of the pressure correction equation are less than a prescribed small value, denoting the convergence criterion. The absolute normalized residual is defined as ... 

Solution of the coupled partial differential and integral equations is performed using a finite difference scheme on an Aspen Custom Modeler platform. Discretization meshes along r and z directions are, respectively, 0.2 mm and 4 mm. To ensure convergence of the numerical scheme, both the fast Newton method for nonlinear solver with convergence criterion on residuals, and a MA48 linear solver are used. Typical simulation duration is 30min on a 3 GHz CPU and 1.5 Go RAM computer with a 10" for the absolute equation tolerance. 

Equation A.2-8 is an absolute convergence criterioru If the value of x changes by less than e from one iteration to the next, the procedure is terminated and the last value of x is taken to be the desired root. The strictness of this criterion for a given e depends on the magnitude of the estimated values. If e = 0.01, for example, and successive estimates of x are 358,234.5 and 358,234.6, the procedure would not be terminated, even though the estimates are undoubtedly close enough for any realistic purpose. On the other hand, successive estimates of 0.0003 and 0.0006 would lead to termination for the same value of e, despite the fact that the two estimates differ by a factor of two. 

The criterion for the convergence of such a sequence is that the absolute eigenvalues of the following Jacobian matrix evaluated at the root are less than 1 ... 

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