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

Abstract

Usual space- and time-staggered (STS) leap-frog Lagrangian hydrodynamic schemes-such as von Neumann-Richtmyer's (1950), Wilkins' (1964), and their variants-are widely used for their simplicity and robustness despite their known lack of exact energy conservation. Since the seminal work of Trulio and Trigger (1950) and despite the later corrections of Burton (1991), it is generally accepted that these schemes cannot be modified to exactly conserve energy while retaining all of the following properties: STS stencil with velocities half-time centered with respect to positions, explicit second-order algorithm (locally implicit for internal energy), and definite positive kinetic energy.It is shown here that it is actually possible to modify the usual STS hydrodynamic schemes in order to be exactly energy-preserving, regardless of the evenness of their time centering assumptions and retaining their simple algorithmic structure. Burton's conservative scheme (1991) is found as a special case of time centering which cancels the term here designated as incompatible displacements residue. In contrast, von Neumann-Richtmyer's original centering can be preserved provided this residue is properly corrected. These two schemes are the only special cases able to capture isentropic flow with a third order entropy error, instead of second order in general.The momentum equation is presently obtained by application of a variational principle to an action integral discretized in both space and time. The internal energy equation follows from the discrete conservation of total energy. Entropy production by artificial dissipation is obtained to second order by a prediction-correction step on the momentum equation.The overall structure of the equations (explicit for momentum, locally implicit for internal energy) remains identical to that of usual STS leap-frog schemes, though complementary terms are required to correct the effects of time-step changes and artificial viscosity updates.In deriving these schemes, an apparently novel approach of flux-in-time was introduced to correct numerical residues and ensure energy conservation. This method can be applied to essentially any numerical scheme whenever required or desired provided space and time numerical consistency is preserved.Numerical test cases are presented confirming the conservative character of the new CSTS schemes down to computer round-off errors, and showing various improvements compared to the standard von Neumann-Richtmyer and Wilkins STS schemes, mostly on shock levels, shock velocities, singularity induced distortions, and CFL stability limits.