Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross‐diffusion system possesses a formal gradient‐flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.     

A finite‐volume scheme for the multidimensional quantum drift‐diffusion model for semiconductors

A finite‐volume scheme for the stationary unipolar quantum drift‐diffusion equations for semiconductors in several space dimensions is analyzed. The model consists of a fourth‐order elliptic equation for the electron density, coupled to the Poisson equation for the electrostatic potential, with mixed Dirichlet‐Neumann boundary conditions. The numerical scheme is based on a Scharfetter‐Gummel type reformulation of the equations. The existence of a sequence of solutions to the discrete problem and its numerical convergence to a solution to the continuous model are shown. Moreover, some numerical examples in two space dimensions are presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1483–1510, 2011     

A finite‐volume scheme for a spinorial matrix drift‐diffusion model for semiconductors

An implicit Euler finite‐volume scheme for a spinorial matrix drift‐diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin‐vector densities, coupled to the Poisson equation for the electric potential. The equations are solved in a bounded domain with mixed Dirichlet–Neumann boundary conditions. The charge and spin‐vector fluxes are approximated by a Scharfetter–Gummel discretization. The main features of the numerical scheme are the preservation of nonnegativity and bounds of the densities and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is unconditionally stable. The fundamental ideas are reformulations using spin‐up and spin‐down densities and certain projections of the spin‐vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic‐layer field‐effect transistor in two space dimensions are presented. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 819–846, 2016     

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.     

Phase-field model for island dynamics in epitaxial growth

A phase-field model is proposed to describe epitaxial growth. In this model the motion of island boundaries of discrete atomic layers is determined by the time evolution of an introduced phase-field variable. We use formally matched asymptotic expansion to determine the asymptotic limit of vanishing interfacial thickness and show the reduction to classical sharp interface models of Burton–Cabrera–Frank type.     

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013     

Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.     

Iterative solution of the drift-diffusion equations

The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical solutions for some coupled systems of nonlinear boundary value problems

Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.     

Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network

In this work, we present and analyze a mathematical model for tumor growth incorporating ECM erosion, interstitial flow, and the effect of vascular flow and nutrient transport. The model is of phase-field or diffused-interface type in which multiple phases of cell species and other constituents are separated by smooth evolving interfaces. The model involves a mesoscale version of Darcy’s law to capture the flow mechanism in the tissue matrix. Modeling flow and transport processes in the vasculature supplying the healthy and cancerous tissue, one-dimensional (1D) equations are considered. Since the models governing the transport and flow processes are defined together with cell species models on a three-dimensional (3D) domain, we obtain a 3D–1D coupled model.     

An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces

We develop an unconditionally stable direct discretization scheme for solving the phase-field crystal equation on surfaces. The surface is discretized by using an unstructured triangular mesh. Gradient, divergence, and Laplacian operators are defined on triangular meshes. The proposed numerical method is second-order accurate in space and time. At each time step, the proposed computational scheme results in linear elliptic equations to be solved, thus it is easy to implement the algorithm. It is proved that the proposed scheme satisfies a discrete energy-dissipation law. Therefore, it is unconditionally stable. A fast and efficient biconjugate gradients stabilized solver is used to solve the resulting discrete system. Numerical experiments are conducted to demonstrate the performance of the proposed algorithm.     

Analysis of a Parabolic Cross-Diffusion Semiconductor Model with Electron-Hole Scattering

The global-in-time existence of non-negative solutions to a parabolic strongly coupled system with mixed Dirichlet–Neumann boundary conditions is shown. The system describes the time evolution of the electron and hole densities in a semiconductor when electron-hole scattering is taken into account. The parabolic equations are coupled to the Poisson equation for the electrostatic potential. Written in the quasi-Fermi potential variables, the diffusion matrix of the parabolic system contains strong cross-diffusion terms and is only positive semi-definite such that the problem is formally of degenerate type. The existence proof is based on the study of a fully discretized version of the system, using a backward Euler scheme and a Galerkin method, on estimates for the free energy, and careful weak compactness arguments.     

On geodesic completeness for Riemannian metrics on smooth probability densities

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are the Otto metric, yielding the \(L^2\)-Wasserstein distance of optimal mass transport, and the Fisher–Rao metric, predominant in the theory of information geometry. On the space of smooth probability densities, none of these Riemannian metrics are geodesically complete—a property desirable for example in imaging applications. That is, the existence interval for solutions to the geodesic flow equations cannot be extended to the whole real line. Here we study a class of Hamilton–Jacobi-like partial differential equations arising as geodesic flow equations for higher-order Sobolev type metrics on the space of smooth probability densities. We give order conditions for global existence and uniqueness, thereby providing geodesic completeness. The system we study is an interesting example of a flow equation with loss of derivatives, which is well-posed in the smooth category, yet non-parabolic and fully non-linear. On a more general note, the paper establishes a link between geometric analysis on the space of probability densities and analysis of Euler–Arnold equations in topological hydrodynamics.     

An Eulerian‐Lagrangian substructuring domain decomposition method for unsteady‐state advection‐diffusion equations

We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001     

Vanishing electron mass limit in the bipolar Euler–Poisson system

The bipolar Euler–Poisson system in physics consists of the conservation laws for the electron and ion densities and their current densities, coupled with the Poisson equation for the electrostatic potential. The limit of vanishing ratio of the electron mass to the ion mass in the $n$-dimensional flat torus is proved in the case of well prepared initial data. The limiting system is composed of two separated equations, where the equation for electron is the incompressible Euler equation with damping, which means physically that the evolution for electrons and ions can be treated as separated motions in the small ratio case.     

An alternating direction implicit scheme for discrete ordinate transport equations

This paper concerns a finite difference approximation of the discrete ordinate equations for the time-dependent linear transport equation posed in a multi-dimensional rectangular parallelepiped with partially reflecting walls. We present an unconditionally stable alternating direction implicit finite difference scheme, show how to solve the difference equations, and establish the following properties of the scheme.If a sequence of difference approximations is considered in which the time and space increments approach zero, then the corresponding sequence of solutions has a subsequence which converges continuously to a strong solution of the discrete ordinate equations. Provided that the time increment is sufficiently small, independently of the space and velocity increment sizes: the solution of the difference equations is bounded by an exponential function of time; in the subcritical case the coefficient of t in this exponential bound is zero or negative; and if the constituent functions are all nonnegative, then the solution of the difference equations will also be nonnegative. This last result implies a monotonicity principle for solutions of related difference problems.     

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.     

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

     

Fourier series and integral equation method for the exterior Stokes problem

The exterior Stokes problem between two parallel planes that are separated by a prismatic cylinder is extended to the interior of the prism by requiring the continuity of the velocity across the lateral faces. The well‐posedness of the exterior–interior problem is proved in suitable weighted Sobolev spaces. The solution is represented by Fourier series in the z‐variable. The Fourier coefficients, solutions of auxiliary two‐dimensional exterior–interior problems, are analyzed by viewing them as boundary integral equations of potential theory and global regularity of the densities, is established in weighted Sobolev spaces of traces. A boundary element method, with suitably refined mesh size, is implemented for the numerical treatment of the Fourier coefficients. This provides optimal convergent semi‐ and fully‐discrete spectral methods of Fourier–Galerkin type. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008