Energy preservation and entropy in Lagrangian space- and time-staggered hydrodynamic schemes

Conservative and entropy controlled remap for multi-material ALE simulations with space-staggered schemes

“…This staggered remapping strategy is then combined with a multi-material conservative space-and time-staggered (CSTS) Lagrangian scheme [24,25] in order to numerically test the proposed strategy. This Lagrangian scheme strictly conserves mass, momentum, angular momentum and total energy [26,27,29]. It also guarantees an increase of entropy and its preservation in an isentropic process up to the scheme's order [24,25,30].…”

“…We now want to couple this general remapping strategy with a specific Lagrangian solver. We focus here on numerical schemes which are space and time staggered [24,29,62] where thermodynamic quantities (material density, material internal energy, material pressure) are defined at cell centers and at time t n while kinematic quantites (velocities) are defined at nodes and calculated at half time t n+ 1 /2 . In what follows, the superscript n + 1 /2 refers to time-staggered quantities while n or n + 1 refer to time-centered ones.…”

A Variational Approach to Design a Numerical Scheme for N-Fluid Flow

“…In some highly demanding fluid dynamics simulations, it appears necessary to simulate multiphase flows involving numerous constraints at the same time, such as: large number of fluids (typically 10 and above), both isentropic and strongly shocked compressible evolution, large heat sources, large deformations, transport over large distances, and highly variable or contrasted equation of state stiffenesses. Fulfilling such a challenge in a robust and tractable way demands that thermodynamic consistency of the numerical scheme be carefully ensured [1,2]. This is addressed here over an arbitrarily evolving computational grid (ALE or Arbitrary Lagrangian-Eulerian approach) by a three-step mimicking derivation [3]: i) to ensure a compatible exchange between kinetic and internal energies under isentropic conditions, a variational least action principle is used to generate the proper pressure forces in the momentum equations; ii) to generate the conservative internal energy equation, a tally is performed to match the kinetic energy, and iii) artificial dissipation is added to ensure shock stability, but other physical terms could also be included (drag, heat exchange, gravity, etc.…”

High-order staggered schemes for compressible hydrodynamics. Weak consistency and numerical validation

“…Before describing our main results for the discretization of system (2), we give an overview of the literature [25] staggered in both time and space whereas Dakin et al proposed conservative and high-order accurate schemes using face-staggering in space and no staggering in time [5]. Our main theoretical results, developed for simplicity for the high-order schemes recently announced in [5], are twofolds.…”

“…Historically the first shock capturing scheme was formulated in internal energy and based on staggered grids [44]. Still, staggered grids schemes are routinely used for CFD applications in industrial context [41,34,16,11,18,25,5]: seminal references are [39,6,7,49]. Two reasons can be argued which are that staggered schemes often need less degrees of freedom than colocated ones to obtain the same accuracy for acoustic propagation [1], and they naturally capture low Mach regimes [11] which is not so easy for standard colocated schemes (see [15] and references therein).…”