Frequency modulation at a moving material interface and a conservation law for wave number

An integral conservation law for wave numbers is considered. In order to test the validity of the proposed conservation law, a complete solution for the reflection and transmission of an acoustic wave impinging normally on a material interface moving at a constant speed is derived. The agreement between the frequency condition thus deduced from the dynamic equations of motion and the frequency condition derived from the jump condition associated with the integral equation supports the proposed law as a true conservation law. Additional comparisons such as amplitude discontinuities and Snells' law in a moving media further confirm the stated proposition. Results are stated concerning frequency and wave number relations across a shock front as predicted by the proposed conservation law.     

On the mathematical conditions for the existence of periodic fluctuations in non-uniform media

In many areas of mathematical physics where one is interested in the propagation of waves through non-uniform media, it is often assumed that periodic excitations result in periodic responses. This assumption is examined by rigorously investigating the existence of periodic solutions of linear hyperbolic differential equations whose coefficients vary with position and whose solution must satisfy periodic boundary or source data. It is shown that the nature of the coefficients of the undifferentiated terms of the differential system is crucial in determining whether or not the solution is periodic. In physical applications, these coefficients usually depend on the gradients of media properties as well as on the media properties themselves. In particular, it is shown that for a general hyperbolic system of two equations in one space dimension, the solution is not periodic. Moreover, this can remain true even if the media gradients are assumed small. However, if the media gradients vanish, or if they vanish except for a bounded region of space, the solution is shown to be periodic for a large enough time. Furthermore, if these gradients vanish asymptotically at large distances, then the disturbances will be asymptotically periodic for increasing time. Special attention is given to the propagation of infinitesimal pressure disturbances through non-uniform steady flows of a lossless fluid.     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Coupled Models and Parallel Simulations for Three-Dimensional Full-Stokes Ice Sheet Modeling

A three-dimensional full-Stokes computational model is considered for determining the dynamics,temperature,and thickness of ice sheets.The goveming thermomechanical equations consist of the three-dimensional full-Stokes system with nonlinear rheology for the momentum,an advective-diffusion energy equation for temperature evolution,and a mass conservation equation for ice-thickness changes.Here,we discuss the variable resolution meshes,the finite element discretizations,and the parallel algorithms employed by the model components.The solvers are integrated through a well-designed coupler for the exchange of parametric data between components.The discretization utilizes high-quality,variable-resolution centroidal Voronoi Delaunay triangulation meshing and existing parallel solvers.We demonstrate the gridding technology,discretization schemes,and the efficiency and scalability of the parallel solvers through computational experiments using both simplified geometries arising from benchmark test problems and a realistic Greenland ice sheet geometry.     

Robin–Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition

Domain decomposition methods for solving the coupled Stokes–Darcy system with the Beavers–Joseph interface condition are proposed and analyzed. Robin boundary conditions are used to decouple the Stokes and Darcy parts of the system. Then, parallel and serial domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence.     

Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations

The mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained.     

Superconvergence analysis of FEMs for the Stokes–Darcy system

We consider a superconvergence analysis for quadratic finite element approximations of the Stokes–Darcy system. The superclose property of an extra half order is proven for uniform triangular meshes. Based on the result of the superclose property, global superconvergence is derived by applying a postprocessing technique. In addition, some numerical examples are presented to demonstrate our theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.     

Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows

A review of adjoint equation-based methodologies for viscous,incompressible flow control and optimization problems is given and illustrated by a drag minimization example. A number of approaches to ameliorating the high storage and CPU costs associated with straightforward implementations of adjoint equation based methodologies are discussed. Other issues, including the relative merits of the differentiate-then-discretize and discretize-then-differentiate approaches to deriving discrete adjoint equations, the incorporation of side constraints into adjoint equation-based methodologies, and inaccuracies that occur due to differentiations at the boundary, are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.