Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: Lessons from the Burgers equation

Finite-dimensional, inviscid equations of hydrodynamics, obtained through a Fourier-Galerkin projection, thermalize with an energy equipartition. Hence, numerical solutions of such inviscid equations, which typically must be Galerkin-truncated, show a behavior at odds with the parent equation. An important consequence of this is an uncertainty in the measurement of the temporal evolution of the distance of the complex singularity from the real domain leading to a lack of a firm conjecture on the finite-time blow-up problem in the incompressible, three-dimensional Euler equation. We now propose, by using the one-dimensional Burgers equation as a testing ground, a numerical recipe, named tyger purging, to arrest the onset of thermalization and hence recover the true dissipative solution. Our method, easily adapted for higher dimensions, provides a tool to not only tackle the celebrated blow-up problem but also to obtain weak and dissipative solutions—conjectured by Onsager and numerically elusive thus far—of the Euler equation.

Figure 1

Representative plots, for $K_{\mathrm{G}}=1000$, of the Galerkin-truncated $v$ (blue) and entropy $u$ (black) solutions of the Burgers equation at (a) $t=0.24\gtrsim t_{*}$ and (b) $t=5.0\gg t_{*}$. For the Galerkin-truncated solution, panel (a) shows signatures of impending thermalization through the birth of tygers while panel (b) shows the fully thermalized solutions. A movie of the time evolution of the Galerkin-truncated equation (and the entropy solution) with a single-mode initial condition for clarity is available [33].

Figure 2

Representative plots, for $K_{\mathrm{G}}=1000$, of the Galerkin-truncated $v$ (blue), the entropy $u$ (black), and the purged $w$ (red) solutions of the Burgers equation at $t=5.0$ for (a) $\alpha =0.6$, $\beta =0.4$ and (b) $\alpha =\beta =0.8$. In panel (b), the purged and entropy solutions are quite close to being identical. A movie of the full evolution in time of the solutions shown in panel (b) is available [34].

Figure 3

(a) A plot of the total energy $E^{\mathrm{P}}$ versus time, from our purged solutions (3), for different combinations of $\alpha$ and $\beta$ and $K_{\mathrm{G}}=1000$. We also show, in black, the energy versus time plot for the entropy solution for comparison. The dashed vertical lines correspond to the times at which the shocks, three in all because of the three-mode initial conditions, form. In the inset, we plot the relative percentage error $e$ (see text) between the purged and entropy solution, for $\alpha =\beta =0.8$, at $t=5.0$, as a function of $K_{\mathrm{G}}$. (b) A plot of the $L_2$ norm of the percentage relative error $\varphi$ (see text) for $\alpha =\beta =0.8$ as a function of $K_{\mathrm{G}}$; the dashed line shows a power-law $K_{\mathrm{G}}^{-1}$ scaling consistent with the measured error.