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Detecting and Preventing Common Errors during Numerical Problem Solving

Mordechai Shacham, Ben Gurion University of the Negev, Chemical Engineering Department, Beer-Sheva, 84105, Israel, Michael B. Cutlip, University of Connecticut, 191 Auditorium Rd., Rm. 204, Storrs, CT 06269-3222, and Michael Elly, Intel Corp., Qiryat Gat, Israel.

            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).