On Convergent Schemes for Two-phase Flow of Dilute Polymeric Solutions

Stefan Metzger

Illinois Institute of Technology

We present a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions, which was derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS, 2016, DOI: 10.1142/S0218202516500196]).

The model consists of a Cahn-Hilliard-Navier-Stokes-Fokker-Planck system describing the evolution of the fluids, the velocity, and the dissolved polymer chains.

We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and thereby establish the convergence of discrete solutions towards a weak solution of the continuous system.

To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

Asymptotic Compatibility of Higher Order Collocation Methods for Nonlocal Problems

Fatih Celiker

Wayne State University

We study convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics.Through a set of numerical experiments we observe that the methods are optimally convergent with order $h^{p+1}$ where $p \ge 0$ is the polynomial degree of the approximation and $h$ is the mesh size. In the case of nonlocal diffusion, it is well known that the solution of the nonlocal problem converges to that of the associated local problem as the horizon parameter $δ > 0$ (measure of nonlocality) tends to zero. A numerical method is said to be asymptotically compatible if the sequence of solutions $u_{δ}^h$ of the nonlocal problem converge to the solution $u_0$ of the local problem as $(δ,h) \to (0,0)$. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for $p \ge 2$, there exists a calibration constant that is independent of $δ$ and $h$ such that the resulting collocation methods for nonlocal diffusion are asymptotically compatible. We verify this finding through extensive numerical experiments.

An Effective Finite Element Iterative Solver for a Poisson-Nernst-Planck Ion Channel Model with Periodic Boundary Conditions

Dexuan Xie

University of Wisconsin-Milwaukee

A system of time-dependent Poisson–Nernst–Planck equations (PNP) is an important dielectric continuum model for simulating ion transport across biological membrane. In this talk, I will report a new PNP ion channel model (PNPic) constructed from periodic boundary value conditions to mimic a membrane environment and to reflect the effects of all the neighboring ion channel proteins. I then will report an effective PNPic finite element solver. One key step of this new solver is to reformulate PNPic into a new system without involving any singular function by solution decomposition techniques. This PNPic solver has been programmed as a software package based on the finite element library from the FEniCs project for an ion channel protein embedded in membrane surrounded by a solvent of anions and cations. Numerical results on a voltage- dependent anion-selective channel protein (PDB ID 3EMN) will be reported to demonstrate the performance of our new software package.

Decoupled, linear, and energy stable finite element method for the Cahn-Hilliard-Navier-Stokes-Darcy phase field model

Xiaoming He

Missouri University of Science and Technology

In this presentation, we discuss an efficient numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn-Hilliard-Navier-Stokes equations in the free flow region and Cahn-Hilliard-Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one only needs to solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

An Efficient Ensemble Algorithm for Numerical Approximation of Stochastic Stokes-Darcy Equations

Changxin Qiu

Missouri University of Science & Technology

We propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic Stokes–Darcy model with a random hydraulic conductivity tensor. The algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately. Moreover, it decouples the Stokes–Darcy system into two smaller sub- physics problems, which reduces the size of the linear systems and allows parallel computation of the two sub-physics problems. We prove the ensemble method is long time stable and first-order in time convergent under a time-step condition and two parameter conditions.

A Kernel-independent Treecode Based on Barycentric Lagrange Interpolation

Lei Wang

University of Wisconsin-Milwaukee

A kernel-independent treecode (KITC) is presented for fast summation of pairwise particle interactions. In general, treecodes replace the particle-particle interactions by particle-cluster interactions, and here we utilize barycentric Lagrange interpolation at Chebyshev points to compute well-separated particle-cluster interactions. The scheme requires only kernel evaluations and is suitable for non-oscillatory kernels. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from $O(N^2)$ to $O(N\ logN)$, where $N$ is the number of particles in the system. The algorithm is demonstrated in serial and parallel simulations for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory overhead. The KITC is a relatively simple algorithm with low memory overhead, and this enables a straightforward efficient parallelization.

A Locally Implicit Lax-Wendroff DG Scheme with Limiters that Guarantee Moment-invertibility for Quadrature Based Moment Closures

Christine Wiersma

Iowa State University

In this work, we consider an approach for approximating kinetic Boltzmann equations known as quadrature-based moment-closures. The true distribution function is replaced by a finite set of Dirac delta functions with variable weights and abscissas. We first show how to construct these Dirac deltas to obtain a set of conservation laws that are conditionally hyperbolic. We then develop a high-order numerical method to discretize the resulting systems, with attention focused on limiters that guarantee that the numerical solutions remain in the convex hyperbolic regions of solution space. Numerical examples will be presented.

Primal-Dual Weak Galerkin Finite Element Methods for Ill-posed Elliptic Cauchy Problems

Chunmei Wang

Texas Tech University

The speaker will present a new numerical method which is devised and analyzed for a type of ill-posed elliptic Cauchy problems by using the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is robust and efficient in the sense that the system arising from the scheme is symmetric, well-posed, and is satisfied by the exact solution (if it exists). The speaker will show some numerical results to demonstrate the efficiency of the primal-dual weak Galerkin method as well as the accuracy of the numerical approximations.