Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

@article{Liu2022PositivitypreservingTO,
  title={Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations},
  author={Hailiang Liu and Zhongming Wang and Peimeng Yin and Hui Yu},
  journal={J. Comput. Phys.},
  year={2022},
  volume={452},
  pages={110777}
}
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A dynamic mass transport method for Poisson-Nernst-Planck equations
. A dynamic mass-transport method is proposed for approximately solving the Poisson–Nernst–Planck (PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally…

References

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Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equations
TLDR
It is proved that the schemes are mass conservative, uniquely solvable and keep positivity unconditionally, and the first-order scheme is proven to be unconditionally energy dissipative.
A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson-Nernst-Planck System
TLDR
This paper constructs an unconditionally stable semi-implicit linearized difference scheme for the time dependent Poisson–Nernst–Planck system, which preserves several important physical laws at full discrete level without any constraints on the time step size.
An Entropy Satisfying Discontinuous Galerkin Method for Nonlinear Fokker–Planck Equations
TLDR
A high order discontinuous Galerkin method for solving nonlinear Fokker–Planck equations with a gradient flow structure 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.
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TLDR
A semi-implicit finite difference scheme for the PNP equations in a bounded domain is introduced and it is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation.
Maximum-Principle-Satisfying Third Order Discontinuous Galerkin Schemes for Fokker-Planck Equations
TLDR
It is shown that a modified limiter can preserve the strict maximum principle for DG schemes solving Fokker--Planck equations, and a scaling limiter for the DG method with first order Euler forward time discretization is proposed to solve the one-dimensional Fokkersonian equations.
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TLDR
This paper designs, analyzes, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem.
Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems
In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations,…
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In these lectures, we will give a general introduction to the discontinuous Galerkin (DG) methods for solving time dependent, convection dominated partial differential equations (PDEs), including the…
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