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6 - Exact solutions

from Part II - Code verification

Published online by Cambridge University Press:  05 March 2013

William L. Oberkampf  and
Christopher J. Roy
Christopher J. Roy
Affiliation:
Virginia Polytechnic Institute and State University
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Summary

The primary focus of this chapter is on the use of exact solutions to mathematical models for code verification. Recall that, in some cases, software testing can be performed by simply running the code and comparing the results to the correct code output. However, in scientific computing, “correct” code output depends on the chosen spatial mesh, time step, iterative convergence tolerance, machine precision, etc. We are thus forced to rely on other less definitive methods for assessing code correctness. In Chapter 5, the order of accuracy test was argued to be the most rigorous approach for code verification. When the order of accuracy test fails, or when the formal order of accuracy has not been determined, then the less rigorous convergence test may be used. In either case, an exact solution to the underlying mathematical model is required. When used for code verification, the ability of this exact solution to exercise all terms in the mathematical model is more important than any physical realism of the solution. In fact, realistic exact solutions are often avoided for code verification due to the presence of singularities and/or discontinuities. Numerous examples will be given in this chapter of exact solutions and their use with the order verification test. The final example given in this chapter employs the less rigorous convergence test with benchmark numerical solutions. In addition to code verification applications, exact solutions to mathematical models are extremely valuable for evaluating the accuracy of numerical schemes, determining solution sensitivity to mesh quality and topology, evaluating the reliability of discretization error estimators, and evaluating solution adaptation schemes. For these secondary applications, physically realistic exact solutions are preferred (see Section 6.4).

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Verification and Validation in Scientific Computing , pp. 208 - 248
Publisher: Cambridge University Press
Print publication year: 2010

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