Structure-preserving and efficient numerical methods for ion transport

Ding, Jie; Wang, Zhongming; Zhou, Shenggao

doi:10.1016/j.jcp.2020.109597

Cited by 24 publications

(21 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recent efforts have been on the design of efficient and stable methods with structurepreserving analysis. On regular domains, results using finite difference/volume for spatial discretization are quite rich, including the works [3,6,7,10,11,13,22,23,24,37], as we discussed above. On irregular domains, Mirzadeh et al [34] presented a conservative hybrid method with adaptive strategies.…”

Section: Further Related Workmentioning

confidence: 99%

“…This example is to test the spatial accuracy of our scheme in a 2D setting. Similar to the Numerical Test 5.1 in [3], we consider the PNP problem (2.1) on Ω = [0, π] 2…”

Section: Numerical Examplesmentioning

confidence: 99%

“…This is evidenced by recent results in [6,10,24] with second order finite difference schemes. Based on some formulations of the nonlogarithmic Landau type (see (1.2) below), the semi-implicit schemes in [3,11,13,22,23] have been shown to feature unconditional positivity, while the energy dissipation is handled differently. For instance, the unconditional positive schemes in [23] are linear, and shown to feature energy dissipation with only an O(1) time step restriction.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Liu,

Wang,

Yin

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Further Related Workmentioning

confidence: 99%

“…This example is to test the spatial accuracy of our scheme in a 2D setting. Similar to the Numerical Test 5.1 in [3], we consider the PNP problem (2.1) on Ω = [0, π] 2…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Liu,

Wang,

Yin

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Convergence analysis and error estimate have been rigorously established for first-order and second-order temporal discretization schemes [19,20]. The other category is based on the Slotboom transformation [7,8,16,21,22,36], which converts the Nernst-Planck (NP) equations into…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, a discrete average of mobility functions is needed at the staggered mesh points, and four different options are considered: the harmonic mean, geometric mean, arithmetic mean, and entropic mean. The mass conservative, positivity-preserving, and energy dissipative properties of the finite difference scheme with various mobility averages have already been established in a few recent works [7,8,21,22]. In particular, the energy dissipation analysis relies on the maximum norm bounds of the concentration, as well as the gradient of the electric potential, while these bounds have to be established through an optimal rate convergence analysis.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson--Nernst--Planck Equations

Ding¹,

Wang²,

Zhou³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The analysis of structure-preserving numerical methods for the Poisson-Nernst-Planck (PNP) system has attracted growing interests in recent years. A class of numerical algorithms have been developed based on the Slotboom reformulation, and the mass conservation, ionic concentration positivity, free-energy dissipation have been proved at a discrete level. Nonetheless, a rigorous convergence analysis for these Slotboom reformulation-based, structure-preserving schemes has been an open problem for a long time. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such results for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.

show abstract

Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

Duan

Chen

Liu

et al. 2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

A class of nonlinear Fokker–Planck equations with nonlocal interactions may include many important cases, such as porous medium equations with external potentials and aggregation–diffusion models. The trajectory equation of the Fokker–Plank equation can be derived based on an energetic variational approach. A structure‐preserving numerical scheme that is mass conservative, energy stable, uniquely solvable, and positivity preserving at a theoretical level has also been designed in the previous work. Moreover, the numerical scheme is shown to satisfy the discrete energetic dissipation law and preserve steady states and has been observed to be second order accurate in space and first‐order accurate time in various numerical experiments. In this work, we give the rigorous convergence analysis for the highly nonlinear numerical scheme. A careful higher order asymptotic expansion is needed to handle the highly nonlinear nature of the trajectory equation. In addition, two step error estimates (a rough estimate and a refined estimate) are necessary in the convergence proof. Different from a standard error estimate, the rough estimate is performed to control the nonlinear term. A few numerical results are also presented to verify the optimal convergence order and the preservation of equilibria.

show abstract

Structure-preserving and efficient numerical methods for ion transport

Cited by 24 publications

References 48 publications

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

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

Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson--Nernst--Planck Equations

Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

Contact Info

Product

Resources

About