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## One Space Dimension (one + space_dimension)
## Selected Abstracts## A space,time discontinuous Galerkin method for the solution of the wave equation in the time domain INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2009Steffen PetersenAbstract In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space,time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave-like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space,time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Some recent finite volume schemes to compute Euler equations using real gas EOS INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002T. GallouëtAbstract This paper deals with the resolution by finite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over ,vacuum', , Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv (,, u, p) and PVRS. The energy relaxation method with VFRoe ncv (,, u, p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for first- and second-order (Runge,Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## One- and Two-Component Bottle-Brush Polymers: Simulations Compared to Theoretical Predictions MACROMOLECULAR THEORY AND SIMULATIONS, Issue 7 2007Hsiao-Ping HsuAbstract Scaling predictions for bottle-brush polymers with a rigid backbone and flexible side chains under good solvent conditions are discussed and their validity is assessed by a comparison with Monte Carlo simulations of a simple lattice model. It is shown that typically only a rather weak stretching of the side chains is realized, and then the scaling predictions are not applicable. Also two-component bottle brush polymers are considered, where two types (A,B) of side chains are grafted, assuming that monomers of different kind repel each other. In this case, variable solvent quality is allowed. Theories predict "Janus cylinder"-type phase separation along the backbone in this case. The Monte Carlo simulations, using the pruned-enriched Rosenbluth method (PERM) give evidence that the phase separation between an A-rich part of the cylindrical molecule and a B-rich part can only occur locally. The correlation length of this microphase separation can be controlled by the solvent quality. This lack of a phase transition is interpreted by an analogy with models for ferromagnets in one space dimension. [source] ## Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Clelia MarchionnaAbstract We introduce a set of conserved quantities of energy-type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well-posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Thermoelasticity with second sound,exponential stability in linear and non-linear 1-d MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2002Reinhard RackeWe consider linear and non-linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave-like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic,parabolic thermoelasticity is given. For Dirichlet type boundary conditions,rigidly clamped, constant temperature,the global existence of small, smooth solutions and the exponential stability are proved for a non-linear system. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Behavior of the solution of a random semilinear heat equation COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2008S. R. S. VaradhanWe consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state. [source] ## Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2004Andrea L. BertozziWe consider a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's "low-curvature image simplifiers" for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k2/(k2 + s2) is a "curvature" threshold and , denotes a fidelity-matching parameter. We derive a priori bounds for ,u that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite-time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second-order Perona-Malik equation (an ill-posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth-order lubrication-type equations and the design of positivity-preserving schemes for such equations. This connection also has relevance for other related fourth-order imaging equations. © 2004 Wiley Periodicals, Inc. [source] ## Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2004Marcello LuciaWe revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linearly unstable equilibrium. For a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension. We introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism. We obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable. Our method also allows us to obtain efficient lower and upper bounds for the propagation speed. © 2004 Wiley Periodicals, Inc. [source] ## Semiclassical limit for the Schrödinger-Poisson equation in a crystal COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2001Philippe BechoucheWe give a mathematically rigorous theory for the limit from a weakly nonlinear Schrödinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant). We introduce a new variant of Wigner transforms, namely the "Wigner Bloch series" as an adaption of the Wigner series for density matrices related to two different "energy bands." Another essential tool are estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian. We assume the initial data to be concentrated in isolated bands but allow for band crossing of the other bands which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation, i.e., we take into account the self-consistent Coulomb interaction. Our results hold also for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors. © 2001 John Wiley & Sons, Inc. [source] |