For simple hydrodynamic solutions, where the pressure and the velocity are polynomial functions of the coordinates, exact microscopic solutions are constructed for the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing and supported by exact boundary schemes. We show how simple numerical and analytical solutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken to adapt them for corners, to examine the uniqueness of the obtained steady solutions and staggered invariants, to validate their exact parametrization by the non-dimensional hydrodynamic and a "kinetic" (collision) number. We also present an inlet/outlet "constant mass flux" condition. We show, both analytically and numerically, that the kinetic boundary schemes may result in the appearance of Knudsen layers which are beyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichlet boundary conditions are investigated for pulsatile flow driven by an oscillating pressure drop or forcing. Analytical approximations are constructed in order to extend the pulsatile solution for compressible regimes.

In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical systems while minimizing computational costs. The algorithm is a generalization of the moving mesh methods based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible, the key is to develop an efficient mesh redistribution procedure so that this part will cost as little as possible comparing with the solution evolution part. Since the mesh redistribution procedure normally requires to solve large size matrix equations, we will describe a procedure to decouple the matrix equation to a much simpler block-tridiagonal type which can be efficiently solved by a particularly designed multi-grid method. To demonstrate the performance of the proposed 3D moving mesh strategy, the algorithm is implemented in finite element simulations of fluid-fluid interface interactions in multiphase flows. To demonstrate the main ideas, we consider the formation of drops by using an energetic variational phase field model which describes the motion of mixtures of two incompressible fluids. Numerical results on two- and three-dimensional simulations will be presented.

The ability to reliably detect coronary artery disease based on the acoustic noises produced by a stenosis can provide a simple, non-invasive technique for diagnosis. Current research exploits the shear wave fields in body tissue to detect and analyze coronary stenoses. The methods and ideas outlined in earlier efforts [6] including a mathematical model utilizing an internal strain variable approximation to the quasi-linear viscoelastic constitutive equation proposed by Fung in [19] is extended here. As an initial investigation, a homogeneous two-dimensional viscoelastic geometry is considered. Being uniform in θ, this geometry behaves as a one dimensional model, and the results generated from it are compared to the one dimensional results from [6]. To allow for different assumptions on the elastic response, several variations of the model are considered. A statistical significance test is employed to determine if the more complex models are significant improvements. After calibrating the model with a comparison to previous findings, more complicated geometries are considered. Simulations involving a heterogeneous geometry with a uniform ring running through the original medium, a θ-dependent model which considers a rigid partial occlusion formed along the inner radius of the geometry, and a model which combines the ring and occlusion are presented.

In this paper we study numerical issues related to the Schrödinger equation with sinusoidal potentials at infinity. An exact absorbing boundary condition in a form of Dirichlet-to-Neumann mapping is derived. This boundary condition is based on an analytical expression of the logarithmic derivative of the Floquet solution to Mathieu's equation, which is completely new to the author's knowledge. The implementation of this exact boundary condition is discussed, and a fast evaluation method is used to reduce the computation burden arising from the involved half-order derivative operator. Some numerical tests are given to show the performance of the proposed absorbing boundary conditions.

The Gierer-Meinhardt reaction-diffusion model is analyzed using a spectral collocation method. This reaction-diffusion system is governed by activator and inhibitor concentrations. Initially, the system is considered in one dimension and then in two dimensions; numerical results are presented for both cases. The algorithmic complexity and accuracy are compared to those of a moving finite element method. Finally, observations are made concerning when to use the proposed spectral method as opposed to the established moving mesh method. 

In this paper, we present an adaptive moving mesh technique for solving the incompressible viscous flows using the vorticity stream-function formulation. The moving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each time step. The Navier-Stokes equations are solved in the vorticity stream-function form by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation in conjunction with a more sophisticated monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. This paper discusses the implementation of the periodic boundary conditions, where the physical domain is allowed to deform with time while the computational domain remains fixed and regular throughout. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.

In this paper, we analyze the stability of the Immersed Boundary Method applied to a membrane-fluid system with a plasma membrane immersed in an incompressible viscous fluid. We show that for small deformations, the planar rest state is stable for a membrane with bending rigidity. The smoothed version, using a standard regularization technique for the singular force, is also shown to be stable. Furthermore, we show that the coupled fluid-membrane system is stiff and smoothing helps to reduce the stiffness. Compared to the system of elastic fibers immersed in an incompressible fluid, membrane with bending rigidity consist of a wider range of decay rates. Therefore numerical instability could occur more easily for an explicit method when the time step size is not sufficiently small, even though the continuous problem is stable.

Due to the complexity of the interactions among the nodes of the complex networks, the properties of the network modules, to a large extent, remain unknown or unexplored. In this paper, we introduce the spatial correlation function G_rs to describe the correlations among the modules of the weighted networks. In order to test the proposed method, we use our method to analyze and discuss the modular structures of the ER random networks, scale-free networks and the Chinese railway network. Rigorous analysis of the existing data shows that the spatial correlation function G_rs is suitable for describing the correlations among different network modules. Remarkably, we find that different networks display different correlations, especially, the correlation function G_rs with different networks meets different degree distribution, such as the linear and exponential distributions.

We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method.