共查询到20条相似文献,搜索用时 15 毫秒
1.
We continue the development of high-order accurate thin layer approximations for time-domain electromagnetics and focus in this paper on a new family of models for thin transmission layers. The thin transmission layer approximations are valid for general isotropic materials and certain types of anisotropic materials. The models also allow the inclusion of smoothly curved layers. Both dielectric and magnetic materials can be considered. These models are non-trivial and we discuss their formulation, properties, and implementation in the context of discontinuous Galerkin methods which emerge as being particularly well suited for this family of models. The range of validity, accuracy, and stability of the models and numerical approximations is demonstrated through one- and two-dimensional examples. 相似文献
2.
The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)2−α+N−r) is proved for the (r−1)th-order finite element method (r≥2). 相似文献
3.
Konstantinos Chrysafinos 《Journal of Computational and Applied Mathematics》2009,231(1):327-348
This work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations. The minimization of the tracking functional subject to implicit parabolic equations is examined. Existence of an optimal solution is proved and an optimality system of equations is derived. Semi-discrete (in space) error estimates for the finite element approximations of the optimality system are presented. These estimates are symmetric and applicable for higher-order discretizations. Finally, fully-discrete error estimates of arbitrarily high-order are presented based on a discontinuous Galerkin (in time) and conforming (in space) scheme. Two examples related to the Lagrangian moving mesh Galerkin formulation for the convection-diffusion equation are described. 相似文献
4.
We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University. 相似文献
5.
We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system
for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that
generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy
inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints
on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver
itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities.
Practical considerations and numerical results will be provided in another paper. 相似文献
6.
Krystyna Twardowska 《Acta Appl Math》1996,43(3):317-359
The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Zong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite- and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the Navier-Stokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.The author's research was partially supported by KBN grant No. 2 P301 052 03. 相似文献
7.
8.
This paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split-step theta method for stochastic systems. First, we consider linear autonomous systems. Under a sufficient and necessary condition for exponential mean square stability of the exact solution, it is proved that the two classes of theta methods with θ≥0.5 are exponentially mean square stable for all positive step sizes and the methods with θ<0.5 are stable for some small step sizes. Then, we study the stability of the methods for nonlinear non-autonomous systems. Under a coupled condition on the drift and diffusion coefficients, it is proved that the split-step theta method with θ>0.5 still unconditionally preserves the exponential mean square stability of the underlying systems, but the stochastic theta method does not have this property. Finally, we consider stochastic differential equations with jumps. Some similar results are derived. 相似文献
9.
Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods. 相似文献
10.
Mingrong Cui 《Journal of Computational and Applied Mathematics》2010,235(3):837-849
High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm. 相似文献
11.
Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and
[16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized
by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which
an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement
indicator into the DLR--Code which is based on a finite volume method of box type on an unstructured grid and present numerical results.
Received May 15, 1995 / Revised version received April 17, 1996 相似文献
12.
Raushan Z. Buzyakova 《Topology and its Applications》2007,154(4):917-924
We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained. 相似文献
13.
A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented. 相似文献
14.
Summary. We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time
evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A
proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known
from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving
a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity
of the free boundary we keep the numerical cost for tracing the free boundary low.
Received June 29, 1998 / Published online June 21, 2000 相似文献
15.
We consider Maxwell’s equations with periodic coefficients as it is usually done for the modeling of photonic crystals. Using Bloch/Floquet theory, the problem reduces in a standard way to a modification of the Maxwell cavity eigenproblem with periodic boundary conditions. Following [8], a modification of edge finite elements is considered for the approximation of the band gap. The method can be used with meshes of tetrahedrons or parallelepipeds. A rigorous analysis of convergence is presented, together with some preliminary numerical results in 2D, which fully confirm the robustness of the method. The analysis uses well established results on the discrete compactness for edge elements, together with new sharper interpolation estimates. 相似文献
16.
An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model. 相似文献
17.
We study an integro-differential equation modeling angular alignment of interacting bundles of cells or filaments. A bifurcation analysis of the related stationary problem was done by Geigant and Stoll in [E. Geigant, M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol. 46 (6) (2003) 537-563]. Here we analyze the time-dependent problem and prove that the type of alignment (one- or multi-directional) depends on the initial distribution, the interaction potential, and the preferred optimal orientation of the bundles of cells or filaments. Our main technical tool is the analysis of the evolution of suitable functionals for the cell density, which allows to also specify the direction(s) where the final alignment takes place. 相似文献
18.
The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method. 相似文献
19.
J. Fontbona 《Probability Theory and Related Fields》2006,136(1):102-156
We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the Lp spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of
the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic
particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic
vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti
in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical Lp spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow
of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three.
Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005. 相似文献
20.
Masaki Hibino 《Journal of Differential Equations》2006,227(2):499-533
This article part I and the forthcoming part II are concerned with the study of the Borel summability of divergent power series solutions for singular first-order linear partial differential equations of nilpotent type. Under one restriction on equations, we can divide them into two classes. In this part I, we deal with the one class and obtain the conditions under which divergent solutions are Borel summable. (The other class will be studied in part II.) In order to assure the Borel summability of divergent solutions, global analytic continuation properties for coefficients are required despite of the fact that the domain of the Borel sum is local. 相似文献