One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes

In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport, proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Communications in Computational Physics. 13(2), 361-385, 2013], completing the mathematical analysis by providing new propositions and new proofs of relations valid across different waves. Next we consider a first order DOT Riemann solver, proposing an integration path that incorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simple well-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, we assess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy.     

Permanent Charge Effects on Ionic Flow: A Numerical Study of Flux Ratios and Their Bifurcation

Ionic flow carries electrical signals for cells to communicate with each other.The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flowproperties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planckmodel for small and relatively large permanent charges. The analytical studies haveled to the introduction of flux ratios that reflect permanent charge effects and have auniversal property. The studies also show that the flux ratios have different behaviorsfor small and large permanent charges. However, the existing analytical techniquescan reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios tobridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. Inparticular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numericaltechniques. An adaptive moving mesh finite element method is used in the numericalstudies.     

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

We propose a coupled model to simulate shallow water waves induced by elastic deformations in the bed topography. The governing equations consist of the depth-averaged shallow water equations including friction terms for the water free-surface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and friction forces are considered for the elastostatic equations whereas bathymetric forces are accounted for in the shallow water equations. The focus in the present study is on the development of a simple and accurate representation of the interaction between water waves and bed deformations in order to simulate practical shallow water flows without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and free-surface waves are investigated in details. Numerical simulations are carried out for several test examples on shallow water waves induced by sudden changes in the bed. The proposed computational model has been found to be feasible and satisfactory.     

Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study

In this paper, we investigate the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparingwith explicit time stepping of the same order accuracy in time, the ETD algorithmscould reduce the computational time in many cases by allowing the use of large timestep sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomainproblems of smaller sizes and solving them locally, the proposed approach could speedup the calculation of matrix exponential vector products. Several standard test casesfor shallow water equations of one or multiple layers are considered. The results showgreat potential of the localized ETD method for high-performance computing becauseeach subdomain problem can be naturally solved in parallel at every time step.     

An Adaptive Moving Mesh Method for the Five-Equation Model

The five-equation model of multi-component flows has been attracting muchattention among researchers during the past twenty years for its potential in the studyof the multi-component flows. In this paper, we employ a second order finite volume method with minmod limiter in spatial discretization, which preserves local extrema of certain physical quantities and is thus capable of simulating challenging testproblems without introducing non-physical oscillations. Moreover, to improve thenumerical resolution of the solutions, the adaptive moving mesh strategy proposedin [Huazhong Tang, Tao Tang, Adaptive mesh methods for one- and two-dimensionalhyperbolic conservation laws, SINUM, 41: 487-515, 2003] is applied. Furthermore, theproposed method can be proved to be capable of preserving the velocity and pressure when they are initially constant, which is essential in material interface capturing.Finally, several classical numerical examples demonstrate the effectiveness and robustness of the proposed method.     

Asymptotic Structure of Cosmological Burgers Flows in One and Two Space Dimensions: A Numerical Study

We study the cosmological Burgers model, as we call it, which is a nonlinearhyperbolic balance law (in one and two spatial variables) posed on an expanding orcontracting background. We design a finite volume scheme that is fourth-order intime and second-order in space, and allows us to compute weak solutions containingshock waves. Our main contribution is the study of the asymptotic structure of thesolutions as the time variable approaches infinity (in the expanding case) or zero (inthe contracting case). We discover that a saddle competition is taking place whichinvolves, on one hand, the geometrical effects of expanding or contracting nature and,on the other hand, the nonlinear interactions between shock waves.     

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradientsmoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization. However, it is not trivial to generalize the vertex-centredEXCMG method to cell-centered finite volume (FV) methods for diffusion equationswith strongly discontinuous and anisotropic coefficients, since a non-nested hierarchyof grid nodes are used in the cell-centered discretization. For cell-centered FV schemes,the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones(primary unknowns). One of the novelties is to propose a new gradient transfer (GT)method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper isto design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStabsmoother for solving the large linear system resulting from linear FV discretizationof diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposedmethod.     

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-Gradient Multi-Component Model with Central Moments Formulation

Multicomponent models based on the Lattice Boltzmann Method (LBM) have clear advantages with respect to other approaches, such as good parallel performances and scalability and the automatic resolution of breakup and coalescence events. Multicomponent flow simulations are useful for a wide range of applications, yet many multicomponent models for LBM are limited in their numerical stability and therefore do not allow exploration of physically relevant low viscosity regimes. Here we perform a quantitative study and validations, varying parameters such as viscosity, droplet radius, domain size and acceleration for stationary and translating droplet simulations for the color-gradient method with central moments (CG-CM) formulation, as this method promises increased numerical stability with respect to the non-CM formulation. We focus on numerical stability and on the effect of decreasing grid-spacing, i.e. increasing resolution, in the extremely low viscosity regime for stationary droplet simulations. The effects of small- and large-scale anisotropy, due to grid-spacing and domain-size, respectively, are investigated for a stationary droplet. The effects on numerical stability of applying a uniform acceleration in one direction on the domain, i.e. on both the droplet and the ambient, is explored into the low viscosity regime, to probe the numerical stability of the method under dynamical conditions.