138 research outputs found
An energy preserving finite difference scheme for the Poisson-Nernst-Planck system
In this paper, we construct a semi-implicit finite difference method for the
time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck
system is a nonlinear system, the numerical method presented in this paper only
needs to solve a linear system at each time step, which can be done very
efficiently. The rigorous proof for the mass conservation and electric
potential energy decay are shown. Moreover, mesh refinement analysis shows that
the method is second order convergent in space and first order convergent in
time. Finally we point out that our method can be easily extended to the case
of multi-ions.Comment: 13 pages, 7 Postscript figures, uses elsart1p.cl
A free energy satisfying finite difference method for Poisson--Nernst--Planck equations
In this work we design and analyze a free energy satisfying finite difference
method for solving Poisson-Nernst-Planck equations in a bounded domain. The
algorithm is of second order in space, with numerical solutions satisfying all
three desired properties: i) mass conservation, ii) positivity preserving, and
iii) free energy satisfying in the sense that these schemes satisfy a discrete
free energy dissipation inequality. These ensure that the computed solution is
a probability density, and the schemes are energy stable and preserve the
equilibrium solutions. Both one and two-dimensional numerical results are
provided to demonstrate the good qualities of the algorithm, as well as effects
of relative size of the data given
Positive and energy stable schemes for Poisson-Nernst-Planck equations and related models
In this thesis, we design, analyze, and numerically validate positive and energy-
dissipating schemes for solving Poisson-Nernst-Planck (PNP) equations and Fokker-Planck (FP) equations with interaction potentials. These equations play an important role in modeling the dynamics of charged particles in semiconductors and biological ion channels, as well as in other applications. These model equations are nonlinear/nonlocal gradient flows in density space, and their explicit solutions are rarely available; however, solutions to such problems feature intrinsic properties such as (i) solution positivity, (ii) mass conservation, and (iii) energy dissipation. These physically relevant properties are highly desirable to be preserved at the discrete level with the least time-step restrictions.
We first construct our schemes for a reduced PNP model, then extend to multi-dimensional PNP equations and a class of FP equations with interaction potentials. The common strategies in the construction of the baseline schemes include two ingredients: (i) reformulation of each underlying model so that the resulting system is more suitable for constructing positive schemes, and (ii) integration of semi-implicit time discretization and central spatial discretization. For each model equation, we show that the semi-discrete schemes (continuous in time) preserve all three solution properties (positivity, mass conservation, and energy dissipation). The fully discrete first order schemes preserve solution positivity and mass conservation for arbitrary time steps. Moreover, there exists a discrete energy function which dissipates along time marching with an bound on time steps.
We show that the second order (in both time and space) schemes preserve solution positivity for suitably small time steps; for larger time steps, we apply a local limiter to restore the solution positivity. We prove that such limiter preserves local mass and does not destroy the approximation accuracy. In addition, the limiter provides a reliable way of restoring solution positivity for other high order conservative finite difference or finite volume schemes.
Both the first and second order schemes are linear and can be efficiently implemented without resorting to any iteration method. The second order schemes are only slight modifications of the first order schemes. Computational costs of a single time step for first and second order schemes are similar, hence our second-order in time schemes are efficient than the first-order in time schemes, given a larger time step could be utilized (to save computational cost). We conduct extensive numerical tests that support our theoretical results and illustrate the accuracy, efficiency, and capacity to preserve the solution properties of our schemes
Energetically stable discretizations for charge carrier transport and electrokinetic models
A finite element discretization using a method of lines approached is
proposed for approximately solving the Poisson-Nernst-Planck (PNP) equations.
This discretization scheme enforces positivity of the computed solutions,
corresponding to particle density functions, and a discrete energy estimate is
established that resembles the familiar energy law for the PNP system. This
energy estimate is extended to finite element solutions to an electrokinetic
model, which couples the PNP system with the Navier-Stokes equations. Numerical
experiments are conducted to validate convergence of the computed solution and
verify the discrete energy estimate.Comment: 22 pages, 2 figure
A Modified Poisson--Nernst--Planck Model with Excluded Volume Effect: Theory and Numerical Implementation
The Poisson--Nernst--Planck (PNP) equations have been widely applied to
describe ionic transport in ion channels, nanofluidic devices, and many
electrochemical systems. Despite their wide applications, the PNP equations
fail in predicting dynamics and equilibrium states of ionic concentrations in
confined environments, due to the ignorance of the excluded volume effect. In
this work, a simple but effective modified PNP (MPNP) model with the excluded
volume effect is derived, based on a modification of diffusion coefficients of
ions. At the steady state, a modified Poisson--Boltzmann (MPB) equation is
obtained with the help of the Lambert-W special function. The existence and
uniqueness of a weak solution to the MPB equation are established. Further
analysis on the limit of weak and strong electrostatic potential leads to two
modified Debye screening lengths, respectively. A numerical scheme that
conserves total ionic concentration and satisfies energy dissipation is
developed for the MPNP model. Numerical analysis is performed to prove that our
scheme respects ionic mass conservation and satisfies a corresponding discrete
free energy dissipation law. Positivity of numerical solutions is also
discussed and numerically investigated. Numerical tests are conducted to
demonstrate that the scheme is of second-order accurate in spatial
discretization and has expected properties. Extensive numerical simulations
reveal that the excluded volume effect has pronounced impacts on the dynamics
of ionic concentration and flux. In addition, the effect of volume exclusion on
the timescales of charge diffusion is systematically investigated by studying
the evolution of free energies and diffuse charges
Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Bikerman Model
We have developed a molecular mean-field theory -- fourth-order
Poisson-Nernst-Planck-Bikerman theory -- for modeling ionic and water flows in
biological ion channels by treating ions and water molecules of any volume and
shape with interstitial voids, polarization of water, and ion-ion and ion-water
correlations. The theory can also be used to study thermodynamic and
electrokinetic properties of electrolyte solutions in batteries, fuel cells,
nanopores, porous media including cement, geothermal brines, the oceanic
system, etc. The theory can compute electric and steric energies from all atoms
in a protein and all ions and water molecules in a channel pore while keeping
electrolyte solutions in the extra- and intracellular baths as a continuum
dielectric medium with complex properties that mimic experimental data. The
theory has been verified with experiments and molecular dynamics data from the
gramicidin A channel, L-type calcium channel, potassium channel, and
sodium/calcium exchanger with real structures from the Protein Data Bank. It
was also verified with the experimental or Monte Carlo data of electric
double-layer differential capacitance and ion activities in aqueous electrolyte
solutions. We give an in-depth review of the literature about the most novel
properties of the theory, namely, Fermi distributions of water and ions as
classical particles with excluded volumes and dynamic correlations that depend
on salt concentration, composition, temperature, pressure, far-field boundary
conditions etc. in a complex and complicated way as reported in a wide range of
experiments. The dynamic correlations are self-consistent output functions from
a fourth-order differential operator that describes ion-ion and ion-water
correlations, the dielectric response (permittivity) of ionic solutions, and
the polarization of water molecules with a single correlation length parameter.Comment: 18 figure
Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations
Keller-Segel systems are a set of nonlinear partial differential equations
used to model chemotaxis in biology. In this paper, we propose two alternating
direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly
with minimal computational cost, while preserving positivity, energy
dissipation law and mass conservation. One scheme unconditionally preserves
positivity, while the other does so conditionally. Both schemes achieve
second-order accuracy in space, with the former being first-order accuracy in
time and the latter second-order accuracy in time. Besides, the former scheme
preserves the energy dissipation law asymptotically. We validate these results
through numerical experiments, and also compare the efficiency of our schemes
with the standard five-point scheme, demonstrating that our approaches
effectively reduce computational costs.Comment: 29 page
An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations
We propose a high order discontinuous Galerkin (DG) method for solving
nonlinear Fokker-Planck equations with a gradient flow structure. For some of
these models it is known that the transient solutions converge to steady-states
when time tends to infinity. The scheme is shown to satisfy a discrete version
of the entropy dissipation law and preserve steady-states, therefore providing
numerical solutions with satisfying long-time behavior. The positivity of
numerical solutions is enforced through a reconstruction algorithm, based on
positive cell averages. For the model with trivial potential, a parameter range
sufficient for positivity preservation is rigorously established. For other
cases, cell averages can be made positive at each time step by tuning the
numerical flux parameters. A selected set of numerical examples is presented to
confirm both the high-order accuracy and the efficiency to capture the
large-time asymptotic
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
Modified Poisson-Nernst-Planck model with Coulomb and hard-sphere correlations
We develop a modified Poisson-Nernst-Planck model which includes both the
long-range Coulomb and short-range hard-sphere correlations in its free energy
functional such that the model can accurately describe the ion transport in
complex environment and under a nanoscale confinement. The Coulomb correlation
{including the dielectric polarization} is treated by solving a generalized
Debye-H\"uckel equation which is a Green's function equation with the
correlation energy of a test ion described by the self Green's function. The
hard-sphere correlation is modeled through the modified fundamental measure
theory. The resulted model is available for problems beyond the mean-field
theory such as problems with variable dielectric media, multivalent ions, and
strong surface charge density. We solve the generalized Debye-H\"uckel equation
by the Wentzel-Kramers-Brillouin approximation, and study the electrolytes
between two parallel dielectric surfaces. In comparison to other modified
models, the new model is shown more accurate in agreement with particle-based
simulations and capture the physical properties of ionic structures near
interfaces.Comment: 22 pages, 7 figure
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