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

Leonardi Filippo

doi:10.3934/dcdss.2018056

Article Contents

2018, Volume 11, Issue 5: 941-961. Doi: 10.3934/dcdss.2018056

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

Leonardi Filippo

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

Received: January 31, 2017

Revised: May 31, 2017

Published: June 2018

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

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Abstract

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.

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Mathematics Subject Classification: 65M12.

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Figure 1. Color map of mean and variance of the vorticity at time $T = 1$ for two different perturbation sizes $\delta$

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Figure 2. $L^2$-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 $\delta = 10e-5$

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Figure 3. Histograms of the vorticity at the point $(0.8,0.8$) and final time $T = 1$ for the perturbed vortex patch

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Figure 4. 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 $512 \times 1024$

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Figure 5. $L^2$-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 $\delta = 10e-4$

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Figure 6. 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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