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

Abstract: Finite-dimensional, inviscid equations of hydrodynamics, such as the zero-viscosity, one-dimensional Burgers equation or the three-dimensional incompressible Euler equation, obtained through a Fourier-Galerkin projection, thermalise---mediated through structures known as tygers [Ray et al., Phys. Rev. E 84, 016301 (2011)]---with an energy equipartition. Therefore, numerical solutions of inviscid partial differential equations, which typically have to be Galerkin-truncated, show a behaviour at odds with the parent equation. We now propose, by using the one-dimensional Burgers equation as a testing ground, a novel numerical recipe, named tyger purging, to arrest the onset of thermalisation and hence recover the true dissipative solution.