Space-Time Adaptive Multiresolution Techniques for Compressible Euler Equations
This paper considers space-time adaptive techniques for finite volume schemes with explicit time discretization. The purpose is to reduce memory and to speed-up computations by a multiresolution representation of the numerical solution on adaptive grids which are introduced by suitable thresholding of its wavelet coefficients. Further speed-up is obtained by the combination of the multiresolution scheme with an adaptive strategy for time integration, which is classical for ODE simulations. It considers variable time steps, controlled by a given precision, using embedded Runge–Kutta schemes. As an alternative to the celebrated CFL condition, the aim in the application of such an time-adaptive scheme for PDE simulations is to obtain accurate and safe integrations. The efficiency of this adaptive space-time method is analyzed in applications to typical Riemann–Lax test problems for the compressible Euler equations in one and two space dimensions. The results show that the accuracy properties of the reference finite volume scheme on the finest regular grid, where the time step is determined by the CFL condition, is preserved. Nevertheless, both CPU time and memory requirements are considerably reduced, thanks to the efficient self-adaptive grid refinement and controlled time-stepping.
KeywordsWavelets Multiresolution Partial differential equations Finite volume Runge–Kutta Adaptivity Time step control
M.O. Domingues and S. Gomes acknowledge financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP grant number: 2012/07281-2, 07/52015-0), and from CNPq (grant number 306828/2010-3, 307511/2010-3, 483226/2011-4)—the Brazilian Research Council, Brazil. They are also grateful for the financial support for visiting positions at École Centrale de Marseille (M.O. Domingues and S.M. Gomes) and Université de Provence (S.M. Gomes). K. Schneider thanks Prof. Carlos de Moura for the invitation to the conference “CFL-Condition: 80 years gone by”, held in Rio de Janeiro in May, 2010. He also acknowledges financial support from the PEPS program of INSMI–CNRS. The authors are grateful to Dominique Fougère, Varlei E. Menconni, and Michel Pognant for their helpful computational assistance.
- 1.Abgrall, R.: Multiresolution analysis on unstructured meshes: applications to CFD. In: Chetverushkin, B.E.A. (ed.) Experimentation, Modelling and Computation in Flow, Turbulence and Combustion. Wiley, New York (1997) Google Scholar
- 16.Kaibara, M., Gomes, S.M.: A fully adaptive multiresolution scheme for shock computations. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications. Kluwer Academic/Plenum Publishers, New York (2001) Google Scholar
- 26.Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Text in Applied Mathematics, vol. 12. Springer, Berlin (1991) Google Scholar