20 research outputs found
Well-balanced finite volume schemes for hydrodynamic equations with general free energy
Well balanced and free energy dissipative first- and second-order accurate
finite volume schemes are proposed for a general class of hydrodynamic systems
with linear and nonlinear damping. The natural Liapunov functional of the
system, given by its free energy, allows for a characterization of the
stationary states by its variation. An analog property at the discrete level
enables us to preserve stationary states at machine precision while keeping the
dissipation of the discrete free energy. These schemes allow for analysing
accurately the stability properties of stationary states in challeging problems
such as: phase transitions in collective behavior, generalized Euler-Poisson
systems in chemotaxis and astrophysics, and models in dynamic density
functional theories; having done a careful validation in a battery of relevant
test cases.Comment: Videos from the simulations of this work are available at
https://sergioperezresearch.wordpress.com/well-balance
Positive and free energy satisfying schemes for diffusion with interaction potentials
In this paper, we design and analyze second order positive and free energy
satisfying schemes for solving diffusion equations with interaction potentials.
The semi-discrete scheme is shown to conserve mass, preserve solution
positivity, and satisfy a discrete free energy dissipation law for nonuniform
meshes. These properties for the fully-discrete scheme (first order in time)
remain preserved without a strict restriction on time steps. For the fully
second order (in both time and space) scheme, we use a local scaling limiter to
restore solution positivity when necessary. It is proved that such limiter does
not destroy the second order accuracy. In addition, these schemes are easy to
implement, and efficient in simulations over long time. Both one and two
dimensional numerical examples are presented to demonstrate the performance of
these schemes.Comment: 29 pages, 3 tables, 6 figure