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Pré-publication, Document de travail

Simulation Verification with PoPe

Abstract : We present the theoretical background of the PoPe and iPoPe verification scheme. The verification that is performed uses the output of actual simulations of production runs. With a small computing overhead it is possible to check that the problem that is solved numerically is consistent with the equations that are to be addressed. In fact, one shows that the numerical error determined by both procedures can be split into a part proportional to the existing operators of the equations, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution can be tested on the error as well as on the modification of the control parameters. To illustrate the method, the evolution equation of a simple mechanical system with two conjugate degrees of freedom is used as simulation test bed. Importantly, although dissipative, the trajectory equations evolve towards a chaotic attractor, a strange attractor, characterised by a positive Lyapunov exponent and therefore sensitivity to initial conditions. It is shown that the chaotic state cannot be verified with the standard Method of Manufactured Solution. We present different facets of the PoPe verification method applied to this test case. We show that the evaluation of the accuracy is case dependent for two reasons. First, the error that is generated depends on the values of the control parameter and not only on the numerical scheme. Second, the target accuracy will depend on the problem one wants to address. In a case characterised by bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one wants to achieve. A unique verification index is proposed to characterise the accuracy, and consequently the verification, of any given simulation in the production runs. This PoPe index then gives a level of confidence of each simulation. A PoPe index of zero characterises a situation with 100% error level. One finds that although the accuracy is poor the robust features of the solution can still be recovered. The maximum PoPe index is determined by machine precision, typically in the range of 12 to 14. As an illustration this PoPe index is used to choose between a high order integration scheme and a reduced order integration scheme that is less precise but requires less operations. For the chosen example the PoPe index indicates that the high order scheme leads to a reduction of computer resources up to a factor 4 at given accuracy.
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https://hal.archives-ouvertes.fr/hal-03531058
Contributeur : Philippe Ghendrih Connectez-vous pour contacter le contributeur
Soumis le : mardi 18 janvier 2022 - 09:20:27
Dernière modification le : vendredi 21 janvier 2022 - 03:45:15

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PoPe_verification_main_part_1....
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  • HAL Id : hal-03531058, version 1

Citation

Philippe Ghendrih, Thomas Cartier-Michaud, Virginie Grandgirard, Eric Serre. Simulation Verification with PoPe. 2022. ⟨hal-03531058⟩

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