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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 4

Special issue on fluid dynamics, analysis and numerics

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Preface
Thomas P. Witelski, David M. Ambrose, Andrea Bertozzi, Anita T. Layton, Zhilin Li and Michael L. Minion
2012, 17(4): i-ii doi: 10.3934/dcdsb.2012.17.4i +[Abstract](414) +[PDF](383.5KB)
Abstract:
Studies of problems in fluid dynamics have spurred research in many areas of mathematics, from rigorous analysis of nonlinear partial differential equations, to numerical analysis, to modeling and applied analysis of related physical systems. This special issue of Discrete and Continuous Dynamical Systems Series B is dedicated to our friend and colleague Tom Beale in recognition of his important contributions to these areas.

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Vorticity jumps in steady water waves
Walter A. Strauss
2012, 17(4): 1101-1112 doi: 10.3934/dcdsb.2012.17.1101 +[Abstract](469) +[PDF](504.5KB)
Abstract:
There exists a large family of water waves with jump discontinuities in the vorticity. These waves travel at a constant speed. They are two-dimensional, periodic, symmetric, and subject to the influence of gravity. Some of them have large amplitudes. Their existence is proven using local and global bifurcation theory, together with elliptic theory of weak solutions with nonlinear boundary conditions.
Well-posedness of a model for water waves with viscosity
David M. Ambrose, Jerry L. Bona and David P. Nicholls
2012, 17(4): 1113-1137 doi: 10.3934/dcdsb.2012.17.1113 +[Abstract](477) +[PDF](418.8KB)
Abstract:
The water wave equations of ideal free-surface fluid mechanics are a fundamental model of open ocean movements with a surprisingly subtle well-posedness theory. In consequence of both theoretical and computational difficulties with the full water wave equations, various asymptotic approximations have been proposed, analyzed and used in practical situations. In this essay, we establish the well-posedness of a model system of water wave equations which is inspired by recent work of Dias, Dyachenko, and Zakharov (Phys. Lett. A, 372:2008). The model in question includes dissipative effects and is weakly nonlinear. The present contribution is a first step in a larger program centered around the Dias-Dychenko-Zhakharov system.
A partially implicit hybrid method for computing interface motion in Stokes flow
Anita T. Layton and J. Thomas Beale
2012, 17(4): 1139-1153 doi: 10.3934/dcdsb.2012.17.1139 +[Abstract](437) +[PDF](249.3KB)
Abstract:
We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. We assume the interface is a closed curve which remains in the interior of the computational region. The implicit time integration is based on the small-scale decomposition approach and does not require the iterative solution of a system of nonlinear equations. First-order and second-order versions of the time-stepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a mesh-based solver. The solutions are second-order accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.
Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces
Xiaohai Wan and Zhilin Li
2012, 17(4): 1155-1174 doi: 10.3934/dcdsb.2012.17.1155 +[Abstract](472) +[PDF](861.7KB)
Abstract:
Solving a Helmholtz equation $\Delta u + \lambda u = f$ efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of $\lambda$ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient $\lambda$ is inversely proportional to the mesh size.
An augmented immersed interface method for moving structures with mass
Jian Hao, Zhilin Li and Sharon R. Lubkin
2012, 17(4): 1175-1184 doi: 10.3934/dcdsb.2012.17.1175 +[Abstract](531) +[PDF](469.1KB)
Abstract:
We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.
Error estimation for immersed interface solutions
Ben A. Vanderlei, Matthew M. Hopkins and Lisa J. Fauci
2012, 17(4): 1185-1203 doi: 10.3934/dcdsb.2012.17.1185 +[Abstract](462) +[PDF](919.9KB)
Abstract:
We present an error estimation method for immersed interface solutions of elliptic boundary value problems. As opposed to an asymptotic rate that indicates how the errors in the numerical method converge to zero, we seek a posteriori estimates of the errors, and their spatial distribution, for a given solution. Our estimate is based upon the classical idea of defect corrections, which requires the application of a higher-order discretization operator to a solution achieved with a lower-order discretization. Our model problem will be an elliptic boundary value problem in which the coefficients are discontinuous across an internal boundary.
Accurate two and three dimensional interpolation for particle mesh calculations
Anita Mayo
2012, 17(4): 1205-1228 doi: 10.3934/dcdsb.2012.17.1205 +[Abstract](457) +[PDF](244.6KB)
Abstract:
We present accurate two and three dimensional methods for interpolating singular or smoothed force fields. The methods are meant to be used in particle mesh or particle-particle particle-mesh calculations so that the resulting schemes conserve momentum. The interpolation weights, which have previously been used by Anderson and Colella to spread charge from particles to the mesh (but not to interpolate the force from the mesh to the particles) use discretizations of the differential equations the forces satisfy. The methods are most accurate when the forces satisfy homogeneous elliptic differential equations or systems of equations, and the precise accuracy levels of the interpolation formulas depend on the accuracy of certain corresponding quadrature formulas. We describe the methods and give results of numerical experiments which demonstrate their effectiveness.
Jet schemes for advection problems
Benjamin Seibold, Rodolfo R. Rosales and Jean-Christophe Nave
2012, 17(4): 1229-1259 doi: 10.3934/dcdsb.2012.17.1229 +[Abstract](695) +[PDF](998.5KB)
Abstract:
We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through the tracking of characteristics and the use of suitable Hermite interpolations, high order is achieved in an optimally local fashion, i.e. the update for the data at any grid point uses information from a single grid cell only. We show that jet schemes can be interpreted as advect-and-project processes in function spaces, where the projection step minimizes a stability functional. Furthermore, this function space framework makes it possible to systematically inherit update rules for the higher derivatives from the ODE solver for the characteristics. Jet schemes of orders up to five are applied in numerical benchmark tests, and systematically compared with classical WENO finite difference schemes. It is observed that jet schemes tend to possess a higher accuracy than WENO schemes of the same order.
Random walks, random flows, and enhanced diffusivity in advection-diffusion equations
Michael Taylor
2012, 17(4): 1261-1287 doi: 10.3934/dcdsb.2012.17.1261 +[Abstract](459) +[PDF](186.7KB)
Abstract:
We study the phenomenon of enhanced diffusivity, introduced by G. I.Taylor, for a class of advection-diffusion equations, modeling, for example, the spread of an ink drop in a fluid engaged in Poiseuille flow. We consider such flow in a pipe of general cross section, and compute variances and covariances of certain random flows associated with the advection-diffusion. We examine both long time behavior, including a central limit theorem, and short time asymptotics.
Simulating binary fluid-surfactant dynamics by a phase field model
Chun-Hao Teng, I-Liang Chern and Ming-Chih Lai
2012, 17(4): 1289-1307 doi: 10.3934/dcdsb.2012.17.1289 +[Abstract](471) +[PDF](2196.0KB)
Abstract:
In this paper, the dynamics of a binary fluid-surfactant system described by a phenomenological phase field model is investigated through analytical and numerical computations. We first consider the case of one-dimensional planar interface and prove the existence of the equilibrium solution. Then we derive the analytical equilibrium solution for the order parameter and the surfactant concentration in a particular case. The results show that the present phase field formulation qualitatively mimics the surfactant adsorption on the binary fluid interfaces. We further study the time-dependent solutions of the system by numerical computations based on the pseudospectral Fourier computational framework. The present numerical results are in a good agreement with the previous theoretical study in the way that the surfactant favors the creation of interfaces and also stabilizes the formation of phase regions.
Asymptotics of blowup solutions for the aggregation equation
Yanghong Huang and Andrea Bertozzi
2012, 17(4): 1309-1331 doi: 10.3934/dcdsb.2012.17.1309 +[Abstract](750) +[PDF](736.0KB)
Abstract:
We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation $ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$, for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$. For $\gamma>2$, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing $\delta$-ring. We develop an asymptotic theory for the approach to this singular solution. For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all $\gamma$ in this range, including additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.
Fundamental limitations of Ad hoc linear and quadratic multi-level regression models for physical systems
Andrew J. Majda and Yuan Yuan
2012, 17(4): 1333-1363 doi: 10.3934/dcdsb.2012.17.1333 +[Abstract](554) +[PDF](767.7KB)
Abstract:
A central issue in contemporary applied mathematics is the development of simpler dynamical models for a reduced subset of variables in complex high dimensional dynamical systems with many spatio-temporal scales. Recently, ad hoc quadratic multi-level regression models have been proposed to provide suitable reduced nonlinear models directly from data. The main results developed here are rigorous theorems demonstrating the non-physical finite time blow-up and large time instability in statistical solutions of general scalar multi-level quadratic regression models with corresponding unphysical features of the invariant measure. Surprising intrinsic model errors due to discrete sampling errors are also shown to occur rigorously even for linear multi-level regression dynamic models. all of these theoretical results are corroborated by numerical experiments with simple models. Single level nonlinear regression strategies with physical cubic damping are shown to have significant skill on the same test problems.

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