Explicit schemes for time propagating many-body wavefunctions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical solution of the time-dependent Schr\"odinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first order differential equations that are characterized by a high degree of stiffness. We analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. We consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness. For the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure

An intelligent processing environment for real-time simulation

The development of a highly efficient and thus truly intelligent processing environment for real-time general purpose simulation of continuous systems is described. Such an environment can be created by mapping the simulation process directly onto the University of Alamba's OPERA architecture. To facilitate this effort, the field of continuous simulation is explored, highlighting areas in which efficiency can be improved. Areas in which parallel processing can be applied are also identified, and several general OPERA type hardware configurations that support improved simulation are investigated. Three direct execution parallel processing environments are introduced, each of which greatly improves efficiency by exploiting distinct areas of the simulation process. These suggested environments are candidate architectures around which a highly intelligent real-time simulation configuration can be developed

Kinematic Dynamos using Constrained Transport with High Order Godunov Schemes and Adaptive Mesh Refinement

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a ``finite-surface'' scheme for the induction equation. This naturally leads to the well-known ``constrained transport'' method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations (referred to as C-MUSCL and U-MUSCL) reach the same level of accuracy as the other one (referred to as Runge-Kutta), at a lower computational cost. More interestingly, these two schemes are compatible with the Adaptive Mesh Refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. We have tested it by solving two kinematic dynamo problems in the low diffusion limit. The construction of this scheme for the induction equation constitutes a step towards solving the full MHD set of equations using an extension of our current methodology.Comment: 40 pages, 10 figures, accepted in Journal of Computational Physics. A version with full resolution is available at http://www.damtp.cam.ac.uk/user/fromang/publi/TFD.pd

The hermite scheme: an application to the n-body problem

In the past century, computational methods have been being applied more and more to physical systems, in special to systems which are chaotic in nature or don’t have an analytical solution, or both, such as is the case for systems that obey the N-body problem. To solve such systems, it is necessary to select the most suitable numerical method, one that takes into account both the necessary time and computational resources available to the researcher, and in order to be able to do so, one must have a good set of tools available. In this work we present a numerical method known as the Hermite Scheme, a fourth-order predictor-corrector integrator which makes use of an Individual Time Step structure, making it capable of processing multi-scale systems. We test its accuracy and study its applicability to the N-body problem, extending the result to chaotic systems in general. We then proceed to check its performance for a N-body system, and compare it to the performance of another fourth-order integrator, the Runge-Kutta. Lastly we verify its performance to multi-scale systems by reproducing some real-life results. Our results show that the Hermite Scheme has a good applicability to N-body systems, with an overall performance better than the fourth order Runge-Kutta. It also shows a good performance when applied to multi-scale systems, with no harm to its overall time performance, which can be applied to other multi-scale systems other than the N-body problem. With this verification, we intend to further apply this method to collision processes and apply the final result on the study of planet formation. The method also shows great potential applicability to Condensed Matter Physics, and we intend to test-apply to known systems in the future.No último século, métodos computacionais vem sendo aplicados mais e mais a problemas físicos, em especial àqueles que ou são caóticos ou não possuem solução analítica, ou ambos, como é o caso de sistemas que obedecem ao problema de N-corpos. Para resolver tais problemas, é necessário selecionar o método numérico mais adequado, um que leve em consideração ambos o tempo necessário e os recursos computacionais disponíveis ao pesquisador responsável; e para que ele seja capaz de fazê-lo, é necessário que ele tenha uma ampla gama de ferramentas disponíveis. Neste trabalho, mostraremos um método numérico conhecido como o Esquema de Hermite, um integrador de quarta ordem preditor- corretor que faz uso de uma estrutura de Passo de Tempo Individual, tornando-o capaz de processar sistemas em multiescalas. Nós testamos sua precisão e estudamos sua aplicação ao problema de N-corpos, estendendo o resultado a sistemas caóticos em geral. Em seguida, verificamos seu desempenho para um sistema de N-corpos e comparamos o resultado com o desempenho de outro integrador de quarta ordem, o Runge-Kutta. Por último nós reproduzimos resultados reais para verificamos seu desempenho em sistemas multiescala. Nossos resultados mostram que o Esquema de Hermite possui uma boa aplicabilidade para sistemas de N-corpos, com um desempenho ao todo melhor do que um Runge-Kutta de quarta ordem. Ele também apresenta um bom desempenho quando aplicado a sistemas multiescala, com nenhum prejuízo à sua performance temporal total, demonstrando que pode ser aplicado a sistemas multiescala que não somente o problema de N-corpos. Com estas verificações, pretendemos no futuro aplicar este método a sistemas com processos de colisão, e aplicar o resultado final no estudo de formação planetária. O método também apresenta grande potencial para aplicação em sistemas de Física da Matéria Condensada, nos quais pretendemos testar a aplicação do método em sistemas conhecidos no futuro.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superio

Assessment of high-order IMEX methods for incompressible flow

This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two third-order RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca $10^{-5}$

Improvement and Application of Smoothed Particle Hydrodynamics in Elastodynamics

This thesis explores the mesh-free numerical method, Smooth Particle Hydrodynamics (SPH), presents improvements to the algorithm and studies its application in solid mechanics problems. The basic concept of the SPH method is introduced and the governing equations are discretised using the SPH method to simulate the elastic solid problems. Special treatments are discussed to improve the stability of the method, such as the treatment for boundary problems, artificial viscosity and tensile instability. In order to improve the stability and efficiency, (i) the classical SPH method has been combined with the Runge-Kutta Chebyshev scheme and (ii) a new time-space Adaptive Smooth Particle Hydrodynamics (ASPH) algorithm has been developed in this thesis. The SPH method employs a purely meshless Lagrangian numerical technique for spatial discretisation of the domain and it avoids many numerical difficulties related to re-meshing in mesh-based methods such as the finite element method. The explicit Runge-Kutta Chebyshev (RKC) scheme is developed to accurately capture the dynamics in elastic materials for the SPH method in the study. Numerical results are presented for several test examples applied by the RKC-SPH method compared with other different time stepping scheme. It is found that the proposed RKC scheme offers a robust and accurate approach for solving elastodynamics using SPH techniques. The new time-space ASPH algorithm which is combining the previous ASPH method and the RKC schemes can achieve not only the adaptivity of the particle distribution during the simulation, but also the adaptivity of the number of stage in one fixed time step. Numerical results are presented for a shock wave propagation problem using the time-space ASPH method compared with the analytical solution and the results of standard SPH. It is found that using the dynamic adaptive particle refinement procedure with adequate refinement criterion, instead of adopting a fine discretisation for the whole domain, can achieve a substantial reduction in memory and computational time, and similar accuracy is achieved

A coupled implicit-explicit time integration method for compressible unsteady flows

This paper addresses how two time integration schemes, the Heun's scheme for explicit time integration and the second-order Crank-Nicolson scheme for implicit time integration, can be coupled spatially. This coupling is the prerequisite to perform a coupled Large Eddy Simulation / Reynolds Averaged Navier-Stokes computation in an industrial context, using the implicit time procedure for the boundary layer (RANS) and the explicit time integration procedure in the LES region. The coupling procedure is designed in order to switch from explicit to implicit time integrations as fast as possible, while maintaining stability. After introducing the different schemes, the paper presents the initial coupling procedure adapted from a published reference and shows that it can amplify some numerical waves. An alternative procedure, studied in a coupled time/space framework, is shown to be stable and with spectral properties in agreement with the requirements of industrial applications. The coupling technique is validated with standard test cases, ranging from one-dimensional to three-dimensional flows

Construction of Low Dissipative High Order Well-Balanced Filter Schemes for Non-Equilibrium Flows

The goal of this paper is to generalize the well-balanced approach for non-equilibrium flow studied by Wang et al. [26] to a class of low dissipative high order shock-capturing filter schemes and to explore more advantages of well-balanced schemes in reacting flows. The class of filter schemes developed by Yee et al. [30], Sjoegreen & Yee [24] and Yee & Sjoegreen [35] consist of two steps, a full time step of spatially high order non-dissipative base scheme and an adaptive nonlinear filter containing shock-capturing dissipation. A good property of the filter scheme is that the base scheme and the filter are stand alone modules in designing. Therefore, the idea of designing a well-balanced filter scheme is straightforward, i.e., choosing a well-balanced base scheme with a well-balanced filter (both with high order). A typical class of these schemes shown in this paper is the high order central difference schemes/predictor-corrector (PC) schemes with a high order well-balanced WENO filter. The new filter scheme with the well-balanced property will gather the features of both filter methods and well-balanced properties: it can preserve certain steady state solutions exactly; it is able to capture small perturbations, e.g., turbulence fluctuations; it adaptively controls numerical dissipation. Thus it shows high accuracy, efficiency and stability in shock/turbulence interactions. Numerical examples containing 1D and 2D smooth problems, 1D stationary contact discontinuity problem and 1D turbulence/shock interactions are included to verify the improved accuracy, in addition to the well-balanced behavior