Spurious vorticity in Eulerian and Lagrangian methods

Lagrangian methods for computational continuum mechanics, since their inception, traditionally relied on staggered meshes. This feature, while facilitating their robustness and reliability, presented some difficulties. The latter motivated the search for collocated Lagrangian schemes. One of the attempts to develop such a scheme was the CAVEAT method/code. Numerical solutions produced by this method suffered sometimes from large vorticity errors, which could lead to mesh entanglement and premature run termination. The efforts to devise a more robust collocated scheme began to bear fruit a couple of decades later starting from the groundbreaking method GLACE, closely followed by EUCCLHYD and later on by CCH and others.

One of the aims of this paper is to present a novel Lagrangian collocated factorizable scheme. The notion of a factorizable method was introduced more than two decades ago within the Eulerian approach. It designates a numerical scheme that reflects/preserves the mixed character of the Euler equations, i.e. does not introduce non-physical coupling between the different factors of the system of equations - advection and acoustics operators.

Another aim of this paper is to explore the connection between the factorizability property of a Lagrangian method and whether or not it suffers from spurious vorticity. Several existing schemes are surveyed for this purpose. A conjecture summarizing our findings is formulated.