The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Implicit Solution of Hyperbolic Equations with Space-Time Adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.     

Quantum dynamics of gas-phase SN2 reactions.

Understanding the state-resolved dynamics of elementary chemical reactions involving polyatomic molecules, such as the well-known reaction mechanism of nucleophilic bimolecular substitution (SN2), is one of the principal goals in chemistry. In this Review, the progress in the quantum mechanical treatment of SN2 reactions in the gas phase is reviewed. The potential energy profile of this class of reactions is characterized by two relatively deep wells, which correspond to pre- and post-reaction chargedipole complexes. As a consequence, the complex-forming reaction is dominated by Feshbach resonances. Calculations in the energetic continuum constitute a major challenge because the high density of resonance states imposes considerable requirements on the convergence and the energetic resolution of the scattering data. However, the effort is rewarding because new insights into the details of multimode quantum dynamics of elementary chemical reactions can be obtained.