2,063 research outputs found
Well-balanced finite volume schemes for nearly steady adiabatic flows
We present well-balanced finite volume schemes designed to approximate the
Euler equations with gravitation. They are based on a novel local steady state
reconstruction. The schemes preserve a discrete equivalent of steady adiabatic
flow, which includes non-hydrostatic equilibria. The proposed method works in
Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any
specific numerical flux and can be combined with any consistent numerical flux
for the Euler equations, which provides great flexibility and simplifies the
integration into any standard finite volume algorithm. Furthermore, the schemes
can cope with general convex equations of state, which is particularly
important in astrophysical applications. Both first- and second-order accurate
versions of the schemes and their extension to several space dimensions are
presented. The superior performance of the well-balanced schemes compared to
standard schemes is demonstrated in a variety of numerical experiments. The
chosen numerical experiments include simple one-dimensional problems in both
Cartesian and spherical geometry, as well as two-dimensional simulations of
stellar accretion in cylindrical geometry with a complex multi-physics equation
of state
Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography
A well-designed numerical method for the shallow water equations (SWE) should
ensure well-balancedness, nonnegativity of water heights, and entropy
stability. For a continuous finite element discretization of a nonlinear
hyperbolic system without source terms, positivity preservation and entropy
stability can be enforced using the framework of algebraic flux correction
(AFC). In this work, we develop a well-balanced AFC scheme for the SWE system
including a topography source term. Our method preserves the lake at rest
equilibrium up to machine precision. The low-order version represents a
generalization of existing finite volume approaches to the finite element
setting. The high-order extension is equipped with a property-preserving flux
limiter. Nonnegativity of water heights is guaranteed under a standard CFL
condition. Moreover, the flux-corrected space discretization satisfies a
semi-discrete entropy inequality. New algorithms are proposed for realistic
simulation of wetting and drying processes. Numerical examples for well-known
benchmarks are presented to evaluate the performance of the scheme
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
Stabilized Lattice Boltzmann-Enskog method for compressible flows and its application to one and two-component fluids in nanochannels
A numerically stable method to solve the discretized Boltzmann-Enskog
equation describing the behavior of non ideal fluids under inhomogeneous
conditions is presented. The algorithm employed uses a Lagrangian
finite-difference scheme for the treatment of the convective term and a forcing
term to account for the molecular repulsion together with a
Bhatnagar-Gross-Krook relaxation term. In order to eliminate the spurious
currents induced by the numerical discretization procedure, we use a
trapezoidal rule for the time integration together with a version of the
two-distribution method of He et al. (J. Comp. Phys 152, 642 (1999)). Numerical
tests show that, in the case of one component fluid in the presence of a
spherical potential well, the proposed method reduces the numerical error by
several orders of magnitude. We conduct another test by considering the flow of
a two component fluid in a channel with a bottleneck and provide information
about the density and velocity field in this structured geometry.Comment: to appear in Physical Review
Numerical methods for all-speed flows for the Euler equations including well-balancing of source terms
This thesis regards the numerical simulation of inviscid compressible ideal gases which are described
by the Euler equations. We propose a novel implicit explicit (IMEX) relaxation scheme to simulate
flows from compressible as well as near incompressible regimes based on a Suliciu-type relaxation
model. The Mach number plays an important role in the design of the scheme, as it has great
influence on the flow behaviour and physical properties of solutions of the Euler equations. Our
focus is on an accurate resolution of the Mach number independent material wave. A special feature
of our scheme is that it can account for the influence of a gravitational field on the fluid flow and is
applicable also in small Froude number regimes. The time step of the IMEX scheme is constrained
only by the eigenvalues of the explicitly treated part and is independent of the Mach number allowing
for large time steps independent of the flow regime. In addition, the scheme is provably asymptotic
preserving and well-balanced for arbitrary a priori known hydrostatic equilibria independently of
the considered Mach and Froude regime. Also, the scheme preserves the positivity of density and
internal energy throughout the simulation, it is well suited for physical applications. To increase the
accuracy, a natural extension to second order is provided. The theoretical properties of the given
schemes are numerically validated by various test cases performed on Cartesian grids in multiple
space dimensions
A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields
This paper extends the gas-kinetic scheme for one-dimensional inviscid
shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to
multidimensional gas dynamic equations under gravitational fields. Four
important issues in the construction of a well-balanced scheme for gas dynamic
equations are addressed. First, the inclusion of the gravitational source term
into the flux function is necessary. Second, to achieve second-order accuracy
of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann
equation with the inclusion of the external force term is used. Third, to avoid
artificial heating in an isolated system under a gravitational field, the
source term treatment inside each cell has to be evaluated consistently with
the flux evaluation at the cell interface. Fourth, the multidimensional
approach with the inclusion of tangential gradients in two-dimensional and
three-dimensional cases becomes important in order to maintain the accuracy of
the scheme. Many numerical examples are used to validate the above issues,
which include the comparison between the solutions from the current scheme and
the Strang splitting method. The methodology developed in this paper can also
be applied to other systems, such as semi-conductor device simulations under
electric fields.Comment: The name of first author was misspelled as C.T.Tian in the published
paper. 35 pages,9 figure
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