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It has been proved inter alia in part I of the present paper(Iserles et al., 1991) that irreducible multistep methods forordinary differential equations may possess period-2 solutionsas asymptotic states if and only if (1)0, where the underlyingmethod is
and $$\sigma \left(z\right):={\sum }_{k=0}^{m}{\sigma }_{\kappa}{z}^{k}$$. We provide an alternative proof of that statementand examine in detail properties of methods that obey (1)=0.By using a variation of the original proof of the first Dahlquistbarrier (Henrici, 1962), we establish an attainable upper boundon the order of zero-stable multistep methods with the aforementionedfeature. Moreover, we modify the concept of backward differentiationformulae (BDF) to require that (1)=0. A zero-stabilitybound on the ensuing methods is produced by extending the methodof proof in (Hairer & Wanner, 1983). 相似文献
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DAVID P. TEW NICHOLAS C. HANDY STUART CARTER STEPHAN IRLE JOEL BOWMAN 《Molecular physics》2013,111(23-24):3513-3525
The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and M?ller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm?1 (obs. 21.56 cm?1), obtained with our quartic density functional surface. 相似文献
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CÉLINE LÉONARD STUART CARTER NICHOLAS C. HANDY PETER J. KNOWLES 《Molecular physics》2013,111(16):1335-1346
The vibrational levels associated with the electronic ground state X2A2″ of NH+ 3 have been determined up to 5000 cm?1 by perturbation and variational calculations with full dimensionality of the molecule. For the variational part a new version of MULTIMODE was used which uses the ab initio electronic energy and its first derivative to define the potential energy function. These quantities were generated by the B97-1 density functional and RCCSD(T) approaches. For ND+ 3, ND2H+ and NDH+ 2 the vibrational levels were calculated only by perturbation theory. The rotational constants for all the isotopomers were determined and the first transition dipole moments for NH+ 3 and ND+ 3 were plotted. A critical comparison of the perturbation and variational techniques suggests a possible further modification to the MULTIMODE algorithm for large systems. 相似文献
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The variational method has been used to determine the geometry and ground state potential surface of acetylene. All the parameters were refined through a least-squares fit to J = 0, 1 levels for C2H2 and C2D2. A new program was written to evaluate the rovibrational energy levels; in particular, primitive basis sets were developed for all values of J taking into account the singularity for linear geometries. Thus Σ, II, δ states can be refined. The full theory for tetraatomic linear molecules is presented. In this refinement 150 observed levels were used as data, below 10 000cm?1. The geometry was refined and gives Re(CC) = 1.2028 Å, Re(CH) = 1.0618 Å, to be compared with the best experimentally derived values of 1.2027 ± 0.0005 Å, 1.062 ± 0.001 Å, respectively. The zero point energies are 5771.1cm?1 for C2H2 and 4571.1cm?1 for C2D2. 相似文献
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Adaptive control of infinite-dimensional systems without parameter estimation: an overview 总被引:1,自引:0,他引:1
LOGEMANN HARTMUT; TOWNLEY STUART 《IMA Journal of Mathematical Control and Information》1997,14(2):175-206
This paper contains an overview of adaptive control of infinite-dimensionalsystems without parameter estimation or identification. We describethe main problems and present the most comprehensive results.High-gain, low-gain and switching controllers are considered.The literature is discussed and a number of open problems areposed. 相似文献
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The existence of solutions of a two-point free-boundary problemarising from the theory of travelling combustion waves in aporous medium is examined. The problem comprises a third-ordernonlinear ordinary differential equation posed on an unknowninterval of finite length; four boundary conditions are given,two at either end of the interval. The equations possess a trivialsolution for all values of the bifurcation parameter . A shootingtechnique is employed to prove the existence of a nontrivialsolution for 0 < < c and nonexistence theorems are provedfor (0, c). 相似文献
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Present address: School of Mathematical Sciences, Bath University, Bath BA2 7AY. A reaction-diffusion equation, coupled through variable heatcapacity and source term to a temporally evolving ordinary differentialequation, is examined. The model is a prototype for the studyof combustion processes where the heat capacity of a compositesolid medium changes significantly as the reactant within themedium is consumed. Similarity solutions are sought by analysing the invarianceof the equations to various stretching groups. The resultingtwo-point boundary-value problem is singular at the origin andposed on the semi-infinite domain. By employing series expansiontechniques we derive a regular problem posed on a finite domain.This problem is amenable to standard numerical solution by meansof Newton-Kantorovich iteration. Results of the computationsare presented and interpreted in terms of the governing partialdifferential equation. 相似文献