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1.
In order to reduce the order of the coefficient vectors of the solutions for the classical Crank–Nicolson collocation spectral (CNCS) method of two-dimensional (2D) viscoelastic wave equations via proper orthogonal decomposition, we first establish a reduced-order extrapolated CNCS (ROECNCS) method of the 2D viscoelastic wave equations so that the ROECNCS method has the same basis functions as the classical CNCS method and maintains all advantages of the classical CNCS method. Then, by means of matrix analysis, we discuss the existence, stability, and convergence for the ROECNCS solutions so that the theory analysis becomes very concise. Finally, we utilize some numerical experiments to validate that the consequences of numerical computations are accordant with the theoretical analysis such that the effectiveness and viability of the ROECNCS method are further verified. Therefore, both theory and method of this paper are new and totally different from the existing reduced-order methods. 相似文献
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We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014 相似文献
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利用特征投影分解(POD)方法建立二维双曲型方程的一种基于POD方法的含有很少自由度但具有足够高精度的降阶有限差分外推迭代格式,给出其基于POD方法的降阶有限差分解的误差估计及基于POD方法的降阶有限差分外推迭代格式的算法实现.用一个数值例子去说明数值计算结果与理论结果相吻合.进一步说明这种基于POD方法的降阶有限差分外推迭代格式对于求解二维双曲方程是可行和有效的. 相似文献
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This paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two-dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper orthogonal decomposition to develop a reduced-order extrapolating STCFE (ROESTCFE) vector model with the lower dimension, and demonstrate the existence, stability, and convergence for the ROESTCFE solutions by the matrix means, resulting in the very concise and flexible theoretical analysis. Lastly, we examine the effectiveness of the developed ROESTCFE model by several numerical tests. It is shown that the ROESTCFE method is computationally very cheap in actual applications. 相似文献
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本文研究无穷凹角区域上一类各向异性问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式和自然积分方程,给出了自然积分方程的数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性. 相似文献
6.
Christophe Audouze Florian De Vuyst Prasanth B. Nair 《Numerical Methods for Partial Differential Equations》2013,29(5):1587-1628
We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
7.
Zhongguo Zhou Jingwen Gu Lin Li Hao Pan Yan Wang 《Numerical Methods for Partial Differential Equations》2021,37(1):479-504
In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L2‐norm. Numerical experiments are presented to illustrate the theoretical analysis. 相似文献
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In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babu?ska for para... 相似文献
10.
In the rectangle D = (0,
,
is considered, where p and
are locally summable functions and may have nonintegrable singularities on . The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbations of the coefficients of the equation under consideration are established. 相似文献
11.
Yunqing Huang Jichun Li Dan Li 《Numerical Methods for Partial Differential Equations》2017,33(3):868-884
In this article, we extend the recently developed weak Galerkin method to solve the second‐order hyperbolic wave equation. Many nice features of the weak Galerkin method have been demonstrated for elliptic, parabolic, and a few other model problems. This is the initial exploration of the weak Galerkin method for solving the wave equation. Here we successfully developed and established the stability and convergence analysis for the weak Galerkin method for solving the wave equation. Numerical experiments further support the theoretical analysis. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 868–884, 2017 相似文献
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Alessandra Aimi Stefano Panizzi 《Numerical Methods for Partial Differential Equations》2014,30(6):2042-2082
A transmission (bidomain) problem for the one‐dimensional Klein–Gordon equation on an unbounded interval is numerically solved by a boundary element method‐finite element method (BEM‐FEM) coupling procedure. We prove stability and convergence of the proposed method by means of energy arguments. Several numerical results are presented, confirming theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2042–2082, 2014 相似文献
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Kun Li Ting-Zhu Huang Liang Li Stéphane Lanteri 《Numerical Methods for Partial Differential Equations》2023,39(2):932-954
In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF2) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems. 相似文献
16.
Zhu Wang 《Numerical Methods for Partial Differential Equations》2015,31(6):1713-1741
For nonlinear reduced‐order models (ROMs), especially for those with high‐order polynomial nonlinearities or nonpolynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. To overcome this issue, we develop an efficient finite element (FE) discretization algorithm for nonlinear ROMs. The proposed approach approximates the nonlinear function by its FE interpolant, which makes the inner product evaluations in generating the nonlinear terms computationally cheaper than that in the standard FE discretization. Due to the separation of spatial and temporal variables in the FE interpolation, the discrete empirical interpolation method (DEIM) can be directly applied on the nonlinear functions in the same manner as that in the finite difference setting. Therefore, the main computational hurdles for applying the DEIM in the FE context are conquered. We also establish a rigorous asymptotic error estimation, which shows that the proposed approach achieves the same accuracy as that of the standard FE method under certain smoothness assumptions of the nonlinear functions. Several numerical tests are presented to validate the proposed method and verify the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1713–1741, 2015 相似文献
17.
S. E. Zhelezovskii 《Siberian Mathematical Journal》2005,46(2):293-304
We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form. 相似文献
18.
Abdollah Shidfar Masoumeh Mohammadi 《Numerical Methods for Partial Differential Equations》2007,23(2):456-474
In this article, we consider a nonlinear partial differential system describing two‐phase transports and try to recover the source term and the nonlinear diffusion term when the state variable is known at different profile times. To this end, we use a POD‐Galerkin procedure in which the proper orthogonal decomposition technique is applied to the ensemble of solutions to derive empirical eigenfunctions. These empirical eigenfunctions are then used as basis functions within a Galerkin method to transform the partial differential equation into a set of ordinary differential equations. Finally, the validation of the used method has been evaluated by some numerical examples. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 456–474, 2007 相似文献
19.
In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first‐order hyperbolic equations are given that describe the evolution of cells and other two second‐order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
20.
This work is concerned mainly with developing and testing the reduced-order extrapolated approach to the unknown coefficient vectors in the Crank-Nicolson finite element (CNFE) solutions for the uniform transmission line equation. For this objective, the CNFE functional and matrix models and the existence, stability, and errors of the CNFE solutions of the uniform transmission line equation are first derived. Then a reduced-order extrapolated CNFE (ROECNFE) matrix model is established by means of a proper orthogonal decomposition technique, and the existence, stability, and error estimates of the ROECNFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Especially, our work shows that the basis functions and accuracy of the ROECNFE matrix model are the same as those of the CNFE matrix model. Finally, some numerical tests are illustrated to computationally experimentally confirm the validity and sharpness of the ROECNFE method. 相似文献