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October 2018, 11(5): 941-961. doi: 10.3934/dcdss.2018056

## A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations

 ETH Zurich, Seminar for Applied Mathematics, Department of Mathematics, HG J 45, Rämistrasse 101, 8092 Zurich, Switzerland

Received  February 2017 Revised  June 2017 Published  June 2018

Fund Project: This work is supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project s590 and by ERC StG NN 306279 SPARCCLE

We formulate a fully discrete finite difference numerical method to approximate the incompressible Euler equations and prove that the sequence generated by the scheme converges to an admissible measure valued solution. The scheme combines an energy conservative flux with a velocity-projection temporal splitting in order to efficiently decouple the advection from the pressure gradient. With the use of robust Monte Carlo approximations, statistical quantities of the approximate solution can be computed. We present numerical results that agree with the theoretical findings obtained for the scheme.

Citation: Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
##### References:
 [1] A. S. Almgren, J. B. Bell and W. G. Szymczak, A numerical method for the incompressible navier--stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17 (1996), 358-369. doi: 10.1137/S1064827593244213. [2] H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [3] S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith and H. Zhang, PETSc Users Manual, Technical Report ANL-95/11 - Revision 3.4, Argonne National Laboratory, 2013. [4] S. Balay, W. D. Gropp, L. C. McInnes and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing (eds. E. Arge, A. M. Bruaset and H. P. Langtangen), Birkhäuser Press, 1997, 163-202. doi: 10.1007/978-1-4612-1986-6_8. [5] C. Bardos and E. Tadmor, Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the 2/3 de-aliasing method, Numerische Mathematik, 129 (2015), 749-782. doi: 10.1007/s00211-014-0652-y. [6] J. B. Bell, P. Colella and H. M. Glaz, A second-order projection method for the incompressible navier-stokes equations, Journal of Computational Physics, 85 (1989), 257-283. doi: 10.1016/0021-9991(89)90151-4. [7] Y. Brenier, C. D. Lellis and L. Székelyhidi Jr, Weak-strong uniqueness for measure-valued solutions, 2009, arXiv: 0912.1028v1. [8] D. Chae, The vanishing viscosity limit of statistical solutions of the Navier-Stokes equations. Ⅱ. The general case, Journal of Mathematical Analysis and Applications, 155 (1991), 460-484. doi: 10.1016/0022-247X(91)90013-P. [9] A. J. Chorin, Numerical solution of the Navier-Stokes Equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [10] J.-M. Delort, Existence de mappes de tourbillon en dimension deux, Journal of the American Mathematical Society, 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6. [11] R. J. DiPerna, Measure valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112. [12] R. J. Diperna and A. Majda, Reduced hausdorff dimension and concentration-cancellation for two dimensional incompressible flow, Journal of the American Mathematical Society, 1 (1988), 59-95. doi: 10.2307/1990967. [13] R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, Communications on Pure and Applied Mathematics, 40 (1987), 301-345. doi: 10.1002/cpa.3160400304. [14] R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Communications in Mathematical Physics, 108 (1987), 667-689. doi: 10.1007/BF01214424. [15] U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math., 17 (2017), 763-827, arXiv: 1402.0909. doi: MR3648106. [16] U. S. Fjordholm, S. Mishra and E. Tadmor, On the computation of measure-valued solutions, Acta Numerica, 25 (2016), 567-679. doi: 10.1017/S0962492916000088. [17] V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [18] R. Glowinski, Finite element methods for incompressible viscous flow, Handbook of numerical analysis, 9 (2003), 3-1176. [19] A. Krzhivitski and O. A. Ladyzhenskaya, A grid method for the Navier-Stokes equations, Soviet Physics Dokl., 11 (1966), 212-213. [20] S. Lanthaler and S. Mishra, Computation of measure-valued solutions for the incompressible Euler equations, 2014, Math. Models Methods Appl. Sci. , 25 (2015), 2043-2088, arXiv: 1411.5064v1. doi: MR3368268. [21] C. Lellis and L. Székelyhidi, On admissibility criteria for weak solutions of the euler equations, Archive for Rational Mechanics and Analysis, 195 (2009), 225-260. doi: 10.1007/s00205-008-0201-x. [22] M. C. Lopes Filho, J. Lowengrub, H. J. Nussenzveig Lopes and Y. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 40 (2006), 225-237. [23] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [24] V. Scheffer, An inviscid flow with compact support in space-time, The Journal of Geometric Analysis, 3 (1993), 343-401. doi: 10.1007/BF02921318. [25] A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Communications on Pure and Applied Mathematics, 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. [26] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. Ⅳ, vol. 39 of Res. Notes in Math., Pitman, Boston, Mass. -London, 1979, 136-212. [27] V. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1456.

show all references

##### References:
 [1] A. S. Almgren, J. B. Bell and W. G. Szymczak, A numerical method for the incompressible navier--stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17 (1996), 358-369. doi: 10.1137/S1064827593244213. [2] H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [3] S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith and H. Zhang, PETSc Users Manual, Technical Report ANL-95/11 - Revision 3.4, Argonne National Laboratory, 2013. [4] S. Balay, W. D. Gropp, L. C. McInnes and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing (eds. E. Arge, A. M. Bruaset and H. P. Langtangen), Birkhäuser Press, 1997, 163-202. doi: 10.1007/978-1-4612-1986-6_8. [5] C. Bardos and E. Tadmor, Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the 2/3 de-aliasing method, Numerische Mathematik, 129 (2015), 749-782. doi: 10.1007/s00211-014-0652-y. [6] J. B. Bell, P. Colella and H. M. Glaz, A second-order projection method for the incompressible navier-stokes equations, Journal of Computational Physics, 85 (1989), 257-283. doi: 10.1016/0021-9991(89)90151-4. [7] Y. Brenier, C. D. Lellis and L. Székelyhidi Jr, Weak-strong uniqueness for measure-valued solutions, 2009, arXiv: 0912.1028v1. [8] D. Chae, The vanishing viscosity limit of statistical solutions of the Navier-Stokes equations. Ⅱ. The general case, Journal of Mathematical Analysis and Applications, 155 (1991), 460-484. doi: 10.1016/0022-247X(91)90013-P. [9] A. J. Chorin, Numerical solution of the Navier-Stokes Equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [10] J.-M. Delort, Existence de mappes de tourbillon en dimension deux, Journal of the American Mathematical Society, 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6. [11] R. J. DiPerna, Measure valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112. [12] R. J. Diperna and A. Majda, Reduced hausdorff dimension and concentration-cancellation for two dimensional incompressible flow, Journal of the American Mathematical Society, 1 (1988), 59-95. doi: 10.2307/1990967. [13] R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, Communications on Pure and Applied Mathematics, 40 (1987), 301-345. doi: 10.1002/cpa.3160400304. [14] R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Communications in Mathematical Physics, 108 (1987), 667-689. doi: 10.1007/BF01214424. [15] U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math., 17 (2017), 763-827, arXiv: 1402.0909. doi: MR3648106. [16] U. S. Fjordholm, S. Mishra and E. Tadmor, On the computation of measure-valued solutions, Acta Numerica, 25 (2016), 567-679. doi: 10.1017/S0962492916000088. [17] V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [18] R. Glowinski, Finite element methods for incompressible viscous flow, Handbook of numerical analysis, 9 (2003), 3-1176. [19] A. Krzhivitski and O. A. Ladyzhenskaya, A grid method for the Navier-Stokes equations, Soviet Physics Dokl., 11 (1966), 212-213. [20] S. Lanthaler and S. Mishra, Computation of measure-valued solutions for the incompressible Euler equations, 2014, Math. Models Methods Appl. Sci. , 25 (2015), 2043-2088, arXiv: 1411.5064v1. doi: MR3368268. [21] C. Lellis and L. Székelyhidi, On admissibility criteria for weak solutions of the euler equations, Archive for Rational Mechanics and Analysis, 195 (2009), 225-260. doi: 10.1007/s00205-008-0201-x. [22] M. C. Lopes Filho, J. Lowengrub, H. J. Nussenzveig Lopes and Y. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 40 (2006), 225-237. [23] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [24] V. Scheffer, An inviscid flow with compact support in space-time, The Journal of Geometric Analysis, 3 (1993), 343-401. doi: 10.1007/BF02921318. [25] A. Shnirelman, On the nonuniqueness of weak solution of the Euler equation, Communications on Pure and Applied Mathematics, 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. [26] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. Ⅳ, vol. 39 of Res. Notes in Math., Pitman, Boston, Mass. -London, 1979, 136-212. [27] V. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics, 3 (1963), 1032-1456.
Color map of mean and variance of the vorticity at time $T = 1$ for two different perturbation sizes $\delta$
$L^2$-norm of the error of the mean and variance of the velocity at time $T = 1$ with different values of $\delta$, for the vortex-patch initial data w.r.t. to a reference solution with $\delta = 10e-5$
Histograms of the vorticity at the point $(0.8,0.8$) and final time $T = 1$ for the perturbed vortex patch
Color map of the mean and the variance of both components of the velocity $\mathbf{u}$, for different values of $\delta$ at time $T = 1$, for a fixed mathcal{G} resolution of $512 \times 1024$
$L^2$-error of the mean and variance for the perturbed vortex-sheet (with root mean square error) as the perturbation size $\delta$ goes to zero. The error is computed against a reference perturbation of size $\delta = 10e-4$
Histograms for the shear-layer at the point $p = (0.5,0.5)$ for the first component of the velocity and for different perturbation sizes (from $\delta = 0.0016$ to $0.0128$) at the time $T = 1$
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