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A second order accurate (in time) numerical scheme is proposed and analyzed for the Poisson–Nernst–Planck equation (PNP) system, reformulated as a non-constant mobility H - 1 gradient flow in the Energetic Variational Approach (EnVarA). The centered finite difference is taken as the spatial discretization. Meanwhile, the highly nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty to design a second order accurate scheme in time, while preserving the variational energetic structures. The mobility function is updated with a second order accurate extrapolation formula, for the sake of unique solvability. A modified Crank–Nicolson scheme is used to approximate the logarithmic term, so that its inner product with the discrete temporal derivative exactly gives the corresponding nonlinear energy difference; henceforth the energy stability is ensured for the logarithmic part. In addition, nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, with the help of the singularity associated with the logarithmic function. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNP system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are also presented. [ABSTRACT FROM AUTHOR]

The well-suited discretization of the Keller–Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller–Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear. [ABSTRACT FROM AUTHOR]

We propose and analyze a first-order in time, second order in space finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states ± 1 , we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values ± 1 in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in l ∞ (0 , T ; L h 2) ∩ l 2 (0 , T ; H h 3) under a linear refinement requirement Δ t ≤ C 1 h . To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states ± 1 , then the numerical solutions can be kept away from ± 1 by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a finite difference numerical scheme for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term, explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection and the phase field coupled term, respectively. Such a semi-implicit method gives an intermediate velocity field. Subsequently, a Helmholtz projection into the divergence-free vector field yields the velocity vector and the pressure variable at the next time step. This approach decouples the Stokes solver, which in turn drastically improves the numerical efficiency. The positivity-preserving property and the unique solvability of the proposed numerical scheme is theoretically justified, i.e., the phase variable is always between -1 and 1, following the singular nature of the logarithmic term as the phase variable approaches the singular limit values. In addition, an iteration construction technique is applied in the positivity-preserving and unique solvability analysis, motivated by the non-symmetric nature of the fluid convection term. The energy stability of the proposed numerical scheme could be derived by a careful estimate. A few numerical results are presented to validate the robustness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H−1 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n and p, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the \ell ^\infty bound for n and p), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme. [ABSTRACT FROM AUTHOR]

Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions. [ABSTRACT FROM AUTHOR]

We present an error analysis for a fully discrete finite difference scheme for the three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn–Hilliard system with a Flory–Huggins–deGennes free energy potential. The numerical scheme was recently proposed, and the positivity-preserving property and unconditional energy stability were theoretically established. In this paper, we rigorously prove first order convergence in time and second order convergence in space for the numerical scheme, in the L Δ t ∞ (0 , T ; H h − 1) ∩ L Δ t 2 (0 , T ; H h 1) norm. Many highly non-standard estimates have to be involved, due to the nonlinear and singular nature of the surface diffusion coefficients. The combination of (i) a higher order asymptotic expansion of the numerical solution (up to second order temporal accuracy); (ii) a rough error estimate (to establish the L Δ t ∞ bound for the phase variables); (iii) and a refined error estimate have to be carried out to conclude such a convergence result. To our knowledge, it will be the first work to provide an optimal rate convergence estimate for a ternary phase field system with singular energy coefficients. [ABSTRACT FROM AUTHOR]

• Convex-concave analysis for the ternary Cahn-Hilliard system with singular potential. • Unique solvability and positivity-preserving analysis for a finite difference scheme. • Unconditional energy stability for such a numerical scheme. • A few interesting numerical simulation results. In this paper, we construct and analyze a uniquely solvable, positivity preserving and unconditionally energy stable finite-difference scheme for the periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. The proposed scheme is based on a convex-concave decomposition of the given energy functional with two variables, and the centered difference method is adopted in space. We provide a theoretical justification that this numerical scheme has a pair of unique solutions, such that the positivity is always preserved for all the singular terms, i.e., not only two phase variables are always between 0 and 1, but also the sum of two phase variables is between 0 and 1, at a point-wise level. In addition, we use the local Newton approximation and multigrid method to solve this nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, energy dissipation and mass conservation properties. [ABSTRACT FROM AUTHOR]

• An energy-dissipation-law-based operator splitting scheme is proposed for reaction-diffusion systems with detailed balance. • The positivity, unique solvability, and energy stability are theoretically justified. • The idea can be applied to various dissipative systems with multiple dissipation components. In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact that both the reaction and diffusion parts dissipate the same free energy opens a path of designing an energy stable, operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme. [ABSTRACT FROM AUTHOR]