Shock Waves

, Volume 29, Issue 1, pp 37–50 | Cite as

On the use of adaptive multiresolution method with time-varying tolerance for compressible fluid flows

  • V. Soni
  • A. HadjadjEmail author
  • O. Roussel
Original Article


In this paper, a fully adaptive multiresolution (MR) finite difference scheme with a time-varying tolerance is developed to study compressible fluid flows containing shock waves in interaction with solid obstacles. To ensure adequate resolution near rigid bodies, the MR algorithm is combined with an immersed boundary method based on a direct-forcing approach in which the solid object is represented by a continuous solid-volume fraction. The resulting algorithm forms an efficient tool capable of solving linear and nonlinear waves on arbitrary geometries. Through a one-dimensional scalar wave equation, the accuracy of the MR computation is, as expected, seen to decrease in time when using a constant MR tolerance considering the accumulation of error. To overcome this problem, a variable tolerance formulation is proposed, which is assessed through a new quality criterion, to ensure a time-convergence solution for a suitable quality resolution. The newly developed algorithm coupled with high-resolution spatial and temporal approximations is successfully applied to shock–bluff body and shock-diffraction problems solving Euler and Navier–Stokes equations. Results show excellent agreement with the available numerical and experimental data, thereby demonstrating the efficiency and the performance of the proposed method.


Multiresolution methods Wavelet adaptive grids Fluid–solid interaction Immersed boundary methods 



This study was supported by the BIOENGINE project, which is funded by the European Regional Development Fund (ERDF) and the Regional Council of Normandie, under contract HN-0002484. The authors also gratefully acknowledge the support of ANR Agence Nationale de la Recherche under grant ANR-13-MONU-0002 (MAPIE project). This work was performed using computing resources from Centre Régional Informatique et d’Applications Numériques de Normandie (CRIANN), Rouen, France.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.INSA of Rouen, CNRS, CORIANormandie UniversityRouenFrance
  2. 2.Cambridge Flow SolutionsCambridgeUK

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