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Register a new userPositivity-preserving third order DG schemes for Poisson--Nernst--Planck equations
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Peimeng Yin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.
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The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport with Faradaic reactions, and have applications in a number of fields. In this article, we develop an adaptive time-stepping scheme for the solution of the PNP-FBV equations based on two time-stepping methods: a fully implicit (BDF2) method, and an implicit-explicit (SBDF2) method. We present simulations under both current and voltage boundary conditions and demonstrate the ability to simulate a large range of parameters, including any value of the singular perturbation parameter $epsilon$. When the underlying dynamics is one that would have the solutions converge to a steady-state solution, we observe that the adaptive time-stepper based on the SBDF2 method produces solutions that ``nearly converge to the steady state and that, simultaneously, the time-step sizes stabilize to a limiting size $dt_infty$. In the companion to this article cite{YPD_Part2}, we linearize the SBDF2 scheme about the steady-state solution and demonstrate that the linearized scheme is conditionally stable. This conditional stability is the cause of the adaptive time-steppers behaviour. While the adaptive time-stepper based on the fully-implicit (BDF2) method is not subject to such time-step constraints, the required nonlinear solve yields run times that are significantly longer.
We present a novel class of high-order space-time finite element schemes for the Poisson-Nernst-Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties. This is accomplished via (1) using finite elements to directly approximate the so-called entropy variable instead of the density variable, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. [17] under the name of a log-density formulation, guarantees both positivity of densities and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. [17].
This work develops entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the pdf of electronic transport along energy bands in semiconductor crystal lattices. We pose, using spherical or energy-angular variables as momentum coordinates, the corresponding Vlasov Boltzmann eq. with a linear collision operator with a singular measure modeling the scattering as functions of the energy band. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position 2D-momentum, and 2D-position 3D-momentum, using the dissipative properties of the collisional operator given its entropy inequality, which depends on the whole Hamiltonian rather than only the kinetic energy. For the 1D problem, knowledge of the analytic solution to Poisson and of the convergence to a constant current is crucial to obtain full stability. For the 2D problem, specular reflection BC are considered in addition to periodicity in the estimate for stability under an entropy norm. Regarding positivity preservation (1D position), we treat the collision operator as a source term and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical pdf at the next time step. The positivity of the numerical pdf in the whole domain is guaranteed by applying the natural limiters that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non-negative. The use of a spherical coordinate system $vec{p}(|vec{p}|,mu=costheta,varphi)$ is slightly different to the choice in previous DG solvers for BP, since the proposed DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms, which is more adequate for Gaussian quadrature than previous approaches.
We design, analyze and numerically validate a novel discontinuous Galerkin method for solving the coagulation-fragmentation equations. The DG discretization is applied to the conservative form of the model, with flux terms evaluated by Gaussian quadrature with $Q=k+1$ quadrature points for polynomials of degree $k$. The positivity of the numerical solution is enforced through a simple scaling limiter based on positive cell averages. The positivity of cell averages is propagated by the time discretization provided a proper time step restriction is imposed.
In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the help of Schauder fixed point theorem, we show the solution existence and uniqueness of the linearized system derived by taking $G-$derivatives of the nonlinear system, followed by the proof of the relationship between the error of solution and the a posteriori error estimator $eta$. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In the further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.
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