(225c) Detecting and Preventing Common Errors during Numerical Problem Solving

            The use of mathematical software packages such as Excel, MATLAB and
POLYMATH for engineering problem solving has brought many benefits including
higher speed and precision in obtaining the results. However, numerical solution
techniques may introduce some new sources of errors. These errors may often pass
undetected as their recognition and correction may require familiarity with some
of the basic concepts of numerical analysis.  Often the study of these concepts
is not included in the typical ChE curriculum.  

            A few examples of errors originating from numerical calculations
follow:

1.  Regression and Analysis of Data

The most common errors in regression of data originate from the using regression
models with too many or too few parameters^[1], fitting polynomial
models without proper scaling (standardization) of the temperature data,
correlation of data when the model equations are improperly linearized^[1],
and regressing data when experimental design for obtaining the data is not
satisfactory^[3].  Sometimes high precision regression models (such as
the Riedel equation for vapor pressure correlation) for which parameters are
available in the literature or databases can also cause significant errors if
the parameter values are carelessly rounded.

2.  Ordinary Differential Equations (ODEs)

Indiscriminate use of default error tolerances of the ODE solver tools is the
most common source of errors in solving ordinary differential equations^[5].
Failure to use the proper integration algorithm (stiff vs. non-stiff),
carelessly rounding numbers in the model equations, using the model outside the
domain of its validity, and the use of low resolution in presenting the results
have been documented^[1],[2] as additional common sources of errors in
solving ODEs.

3.  Systems of Nonlinear Algebraic Equations (NLEs)

There are many examples in the literature^[4] showing that identifying
initial guesses and formulating the problem correctly to enable convergence to a
solution of an NLE system represent major challenges.  Even if the solution is
found, it may be infeasible in the physical sense (solution with negative
concentrations, for example) or a false solution caused by improper variable
and/or function scaling.

            It is important that students become able to recognize an erroneous
solution and then make the necessary corrections.  This may require familiarity
with concepts associated with numerical analysis such as "ill-conditioned
matrices", "stiff ODEs", "round of errors in computation and error propagation"
and "radius of convergence." Thus the ChE curriculum should include these topics
either as the numerical problem solving is introduced to the student (integrated
throughout the curriculum) or in a required "numerical methods" course.

            This paper will demonstrate several examples typical errors
introduced into numerical calculations and suggest the needed corrections.  The
recommended structure of a "Process Modeling and Numerical Methods" course will
also be described.   

References

1. N. Brauner, M. Shacham and M. B. Cutlip, ?Computational Results: How Reliable Are They? A Systematic Approach to Modal Validation,'' Chem. Eng. Educ., 30 (1), 20-25 (1996).
2. Shacham, M., N. Brauner and M. Pozin, "Potential  Pitfalls in Using General Purpose Software for Interactive Solution of Ordinary Differential Equations," Acta Chimica Slovenica, 42(1), 119-124 (1995) .
3. Shacham, M. and N. Brauner, "Correlation and Over-correlation of Heterogeneous Reaction Rate Data," Chem. Eng. Educ., 29(1) 22-25, 45 (1995).
4. Shacham, M., N. Brauner and M. B. Cutlip, ?A Web-based Library for Testing Performance of Numerical Software for Solving Nonlinear Algebraic Equations,? Computers Chem. Engng. 26(4-5), 547-554(2002).
5. Shacham, M., N. Brauner, W. R. Ashurst and M. B. Cutlip, "Can I Trust this Software Package? ? An Exercise in Validation of Computational Results," Chem. Eng. Educ., 42(1), 53-59 (2008).