What is numerical analysis?
Numerical Analysis is a crossroads of disciplines, which makes both its interest and difficulty. The study and resolution of a full scale numerical problem goes through several phases:
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First, the problem comes generally, from mechanics, physics, engineering sciences,.... There is thus a preliminary work of modeling, resulting in a setting into equations. Generally this modeling is nontrivial. It is made, or at least engaged by who poses the problem (physicist, mechanics, engineer,...).
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The preceding stage leads to a mathematical model. Here comes the mathematician. He proposes an adequate functional framework and endeavors to show that the problem arising, or at least its mathematical translation, admits a single solution. Unfortunately, in many real situations, this study cannot be concluded, either because of the mathematical difficulty of the task or because of lack of time and means. Concerning the mathematical difficulty, let us recall that the mathematical study of certain equations of fluid mechanics, nowadays known for several centuries, is far from being completed.
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The following step consists in defining an approximation of the model, in order to allow a numerical resolution of the problem. There begins the work of the numerical analyst. It is necessary for him to build a computation algorithm and, when possible, to show that this algorithm defines an approximate solution of the problem.
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Then, it remains to write a software implementing the algorithm. Last but not least, it remains to validate the results provided at the end of the chain by the machine.
Throughout this process, the sources of error are multiple:
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Did the mathematical model take into account the essential physical phenomena, neglecting only what is negligible?
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Is the built model mathematically coherent? Does it define one only one solution?
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Is the obtained approximation a good one? Which is the sensitivity of the model to the errors? Can small errors in the data generate big errors in the computed solution?
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Is the resulting software correct? In a software of several tens, even hundreds of thousands of instructions, there are always programming errors.
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Finally each arithmetic operation is carried out with a rounding error. When a super calculator carries out milliards at the second and runs, say twenty minutes, one can legitimately fear that this leads to a catastrophic accumulation, sufficient in itself to remove any validity of the results.
Thus can be seen the absolute necessity of a validation of the results. It can also be seen that one will never have certainty, but only at best a beam of presumptions of validity. The methods to be used can be classified summarily in the following way:
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Study of the qualitative properties of the calculated solution and its physical probability. Here in an essential way, the physicist (engineer, chemist,...) who posed the problem will intervene. He will be able to say to what the sought physical solution should look like (it is known for example, that the mechanical constraints in a solid become very large close to the edges).
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Calculation of analytical solutions, in particular cases. Before the advent of compute, a whole arsenal of special functions (functions of Bessel, spherical harmonics, orthogonal polynomials,...) had been built in order to express the solutions of problems in the case of simple geometries. This arsenal is invaluable and must be used each time it is possible.
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One can often give oneself an arbitrary solution, then build the corresponding data, to launch then the software on this data, to see whether the computed solution coincides with the solution we started from.