共查询到20条相似文献,搜索用时 21 毫秒
1.
This paper is concerned with a heat diffusion problem in a half-space which is motivated by the detection of material defects
using thermal measurements. This problem is solved by inverting the Laplace transform with respect to time on a contour in
the complex plane using an exponentially convergent quadrature rule. This leads to a finite number of time-independent problems,
which can be solved in parallel using boundary integral equation methods. We provide a full numerical analysis of this scheme
on compact time intervals. Our results are formulated in a way that they can easily be used for other diffusion problems in
exterior or interior domains. 相似文献
2.
This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu
t
+uu
x
−Hu
xx
=0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for
a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations
in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula
which may be computed efficiently using the Fast Fourier Transform. 相似文献
3.
We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases.
A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based
on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements
are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370,
2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization
parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas
elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples. 相似文献
4.
Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary. 相似文献
5.
Da Xu 《Journal of Computational and Applied Mathematics》2011,236(5):684-698
We consider order one operational quadrature methods on a certain integro-differential equation of Volterra type on (0,∞), with piecewise linear convolution kernels. The forms of discretization solution are patterned after a continuous one of Hannsgen (1979) [2]. An l1 remainder stability and an error bound are derived. 相似文献
6.
A. Ayd?nB. Karasözen 《Journal of Computational and Applied Mathematics》2011,235(16):4770-4779
In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass. 相似文献
7.
Zhiyong Si Yinnian He Tong Zhang 《Journal of Computational and Applied Mathematics》2012,236(9):2553-2573
In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems. 相似文献
8.
This paper considers the spectral methods for a Volterra-type integro-differential equation. Firstly, the Volterra-type integro-differential equation is equivalently restated as two integral equations of the second kind. Secondly, a Legendre-collocation method is used to solve them. Then the error analysis is conducted based on the L∞-norm. In addition, numerical results are presented to confirm our analysis. 相似文献
9.
Viktoriya Masol Jef L. Teugels 《Journal of Computational and Applied Mathematics》2010,233(10):2521-1054
In this article, we investigate and compare a number of real inversion formulas for the Laplace transform. The focus is on the accuracy and applicability of the formulas for numerical inversion. In this contribution, we study the performance of the formulas for measures concentrated on a positive half-line to continue with measures on an arbitrary half-line. As our trial measure concentrated on a positive half-line, we take the broad Gamma probability distribution family. 相似文献
10.
Rico Zacher 《Journal of Evolution Equations》2005,5(1):79-103
We prove maximal regularity results of type Lp for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on Lp-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition. 相似文献
11.
12.
We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France. 相似文献
13.
14.
15.
Goro Akagi Kazumasa Suzuki 《Calculus of Variations and Partial Differential Equations》2008,31(4):457-471
The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate
nonlinear parabolic equation of the form , where denotes the so-called infinity-Laplacian given by . To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify
the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results
on the Cauchy–Dirichlet problem.
Dedicated to the memory of our friend Kyoji Takaichi.
The research of the first author was partially supported by Waseda University Grant for Special Research Projects, #2004A-366. 相似文献
16.
We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with
a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular
behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the
error is of order k
2 + h
2, where k and h are the parameters for the time and space meshes, respectively. 相似文献
17.
In this paper we study how to find solutions to the parabolic Ginzburg–Landau equation that as have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a
uniform profile for the solution up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant. 相似文献
18.
Following earlier work by Sheen, Sloan, and Thomée concerningparabolic equations we study the discretization in time of aVolterra type integro-differential equation in which the integraloperator is a convolution of a weakly singular function andan elliptic differential operator in space. The time discretizationis accomplished by using a modified Laplace transform in timeto represent the solution as an integral along a smooth curveextending into the left half of the complex plane, which isthen evaluated by quadrature. This reduces the problem to afinite set of elliptic equations with complex coefficients,which may be solved in parallel. Stability and error boundsof high order are derived for two different choices of the quadraturerule. The method is combined with finite-element discretizationin the spatial variables. 相似文献
19.
N. Yu. Bakaev 《BIT Numerical Mathematics》1997,37(2):237-255
The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential
multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev
inequality and some operator representations of the finite element solutions. The results of the present paper lead to the
error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements. 相似文献
20.
The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated. 相似文献