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991.
Bülent Karaszen Süleyman Yldz Murat Uzunca 《Mathematical Methods in the Applied Sciences》2021,44(1):476-492
In this paper, we present two different approaches for constructing reduced‐order models (ROMs) for the two‐dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear‐quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem. 相似文献
992.
Boniface Nkemzi Michael Jung 《Numerical Methods for Partial Differential Equations》2021,37(1):836-853
Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. 相似文献
993.
Kolade M. Owolabi 《Numerical Methods for Partial Differential Equations》2021,37(1):131-151
This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices. 相似文献
994.
We present a new algorithm for generating layer-adapted meshes for the
finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar
to the well-known Bakhvalov meshes, but can be used in more general settings:
specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms,
and numerical verification of their properties in a variety of settings. The MPDE is
a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh
intervals. We resolve this by proposing a scheme based on $h$-refinement. We present
fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)]
that implement these methods, facilitating their extension to other problems and
settings. 相似文献
995.
Yongtao LIU 《Frontiers of Mathematics in China》2021,16(3):783
By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of (K,N). This generalizes a main result of F. Münch who considers the case of CD(K, ) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian and perpetual cutoff semigroup in our setting. 相似文献
996.
JOSE PITARCH-RUIZ JOSE SANCHEZ-MARIN ALFREDO SANCHEZ DE MERAS DANIEL MAYNAU 《Molecular physics》2013,111(3):483-494
The vertical electronic spectrum of formaldehyde has been studied by means of (SC)2-MR-SDCI and CCLR methods. Two basis sets of atomic natural orbitals (ANOs) complemented with a one-centre series of Rydberg orbitals were used. The first was taken from the CASPT2 study by Merchán, M., and Roos, B. O., 1995, Theoret. Chim. Acta, 92, 221, and may be described as C,O[4s3pld]/H[2slp] with a lslpld Rydberg series centred in the charge centroid of the 2B2 state of the cation. The second was a larger basis set that may be described as C,O[6s5p3d2f]/H[4s3p2d] + 3s3p3d in the same centre. The (SC)2 dressing may be applied efficiently to an MR-SDCI method and comparison with the dressed CAS-SDCI is satisfactory, in spite of the remarkable reduction in the CI space dimension. The consistency of the (SC)2-MR-SDCI results was tested also against the CCLR and CASPT2 results using the same basis sets and against the CCLR results using Dunning's aug- and daug-cc-pVQZ basis sets. The 3A1(π → π *) state is correctly placed as the second excited triplet while the highly multi-configurational nature of the 1A1(π → π *) state is confirmed as well as its greatly mixed valence-Rydberg nature. This state is predicted as lying under the 10 eV level, on top of the (ny → 3d) Rydberg states that are predicted in the 8.9–9.5eV region. The 5 1B2(ny → 4s) Rydberg state and the 1B2(σy → π*) also are predicted in this region. The triplet states also were calculated with the (SC)2-MR-SDCI method. The vertical ordering of the 2 1A1(ny → 3py) and 2 1B2(ny → 3pz) states is discussed, as well as that of the 1B1(σ → π*) and the Rydberg 1B1(ny → 3dxy) states. This work shows the highly reliable values that may be reached applying the dressing method along with a large basis set. Such a procedure is made possible using an MR-SDCI selection of spaces instead of the CAS-SDCI that was used up to now in most (SC)2 dressing applications. 相似文献
997.
Zbigniew Romanowski 《Molecular physics》2013,111(13):1339-1348
The h-adaptive, high order finite element method is applied to solve a second order one dimension eigenvalue problem. The finite element formulation for the Lobatto basis is given, for which basis functions of arbitrary order can be constructed. The adaptive algorithm is simple, yet very efficient and straightforward to implement. The algorithm is based on the observation that the expansion coefficients of Lobatto basis functions decay rapidly. It allows evaluating the smallest eigenvalues simultaneously with the comparable accuracy for all eigenvalues. The presented algorithm is applied to solve the radial Schrödinger equation with the Coulomb and the Woods–Saxon potentials. For both potentials the convergence rate is presented. After seven adaptive iterations nine-digit accuracy was obtained. 相似文献
998.
Semiconductor diamond is considered the best heater material to generate ultra-high temperatures in a Kawai cell. In two pioneering studies, a mixture of graphite and amorphous boron (or boron carbide, B4C) was converted to semiconductor diamond in the diamond stability field and was confirmed to generate 2000°C and 3500°C, respectively. Following these works, we synthesized a homemade boron-doped graphite block with fine machinability. With this technical breakthrough, we developed a semiconductor diamond heater in a smaller Kawai-type cell assembly. Here, we report the procedure for making machinable boron-doped graphite, and the performance of the material as a heater in a Kawai cell at 15?GPa using tungsten carbide anvils and at ~50?GPa using sintered diamond anvils. Furthermore, we present a finite element simulation of the temperature distribution generated by a semiconductor diamond heater, which is much more homogeneous than that generated by a metal heater. 相似文献
999.
《Composite Interfaces》2013,20(6):507-520
The aim of this article is to provide a systematic method for performing experimental tests and theoretical evaluations on interfacial adhesion properties of the W/Al bilayer thin films interface. Samples W/Al bilayer thin films assembly is deposited on the quartz glass by using radio frequency magnetron sputtering. Based on the analysis of the experimental indentation data, the elastic modulus and hardness of the sample are investigated. The test results show that both of the values are easily influenced by the indentation depth. At the meantime, a finite element model is built to simulate the interface mechanical properties. The analysis shows that stress is mainly centralized close to the indenter and the maximum stress occurs in the lower layer Al film, not in the upper W film. The comparison between the experiment and the simulation shows the validity of the test and the modeling of each other to a certain extent. The investigation builds a basis for future work such as the fabrication of W/Al bilayer thin films for micro/nano manufacturing. 相似文献
1000.
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