EFFICIENT NUMERICAL INTEGRATORS FOR HIGHLY OSCILLATORY DYNAMIC SYSTEMS BASED ON MODIFIED MAGNUS INTEGRATOR METHOD

Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the secondorder dynamic system was reformulated as the first-order system and the frame of reference was transfered by introducing new variables so that highly oscillatory behaviour inherits from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately designed for solving the above new form and some improved also were presented. Finally, numerical examples show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness.

LONG-TIME OSCILLATORY ENERGY CONSERVATION OF TOTAL ENERGY-PRESERVING METHODS FOR HIGHLY OSCILLATORY HAMILTONIAN SYSTEMS

For an integrator when applied to a highly oscillatory system,the near conservation of the oscillatory energy over long times is an important aspect.In this paper,we study the long-time near conservation of oscillatory energy for the adapted average vector field(AAVF)method when applied to highly oscillatory Hamiltonian systems.This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly.This paper is devoted to analysing another important property of AAVF method,i.e.,the near conservation of its oscillatory energy in a long term.The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion.A similar result of the method in the multi-frequency case is also presented in this paper.

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.

Levin methods for highly oscillatory integrals with singularities

In this paper,new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities.To avoid singularities,the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs.The solutions of one can be obtained explicitly,while the other kind of ODEs can be solved efficiently by collocation methods.The proposed methods can attain arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points.Several numerical experiments are presented to validate the efficiency of the proposed methods.

Long Time Energy and Kinetic Energy Conservations of Exponential Integrators for Highly Oscillatory Conservative Systems

In this paper,we investigate the long-time near-conservations of energy and kinetic energy by the widely used exponential integrators to highly oscillatory conservative systems.The modulated Fourier expansions of two kinds of exponential integrators have been constructed and the long-time numerical conservations of energy and kinetic energy are obtained by deriving two almost-invariants of the expansions.Practical examples of the methods are given and the theoretical results are confirmed and demonstrated by a numerical experiment.

HETEROGENEOUS MULTISCALE METHOD FOR OPTIMAL CONTROL PROBLEM GOVERNED BY ELLIPTIC EQUATIONS WITH HIGHLY OSCILLATORY COEFFICIENTS

In this paper, we investigate heterogeneous multiscale method （HMM） for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well- known Lions＇ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both L^2 and H^1 norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

Trigonometric WENO Schemes for Hyperbolic Conservation Laws and Highly Oscillatory Problems

In this paper,we use trigonometric polynomial reconstruction,instead of algebraic polynomial reconstruction,as building blocks for the weighted essentially non-oscillatory(WENO)finite difference schemes to solve hyperbolic conservation laws and highly oscillatory problems.The goal is to obtain robust and high order accurate solutions in smooth regions,and sharp and non-oscillatory shock transitions.Numerical results are provided to illustrate the behavior of the proposed schemes.

MULTISCALE ASYMPTOTIC EXPANSION FOR A CLASS OF HYPERBOLIC-PARABOLIC TYPE EQUATION WITH HIGHLY OSCILLATORY COEFFICIENTS

In this paper, we will discuss the asymptotic behaviour for a class of hyperbolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of composite media. A complete multiscale asymptotic expansion and its rigorous verification will be reported.

Effect of High Frequency Oscillatory Ventilation on EVLW and Lung Capillary Permeability of Piglets with Acute Respiratory Distress Syndrome Caused by Pulmonary and Extrapulmonary Insults

The effect of high frequency oscillatory ventilation（HFOV） at early stage on hemodynamic parameters, extravascular lung water（EVLW）, lung capillary permeability, CC16 and s ICAM-1 in piglets with pulmonary or extrapulmonary acute respiratory distress syndrome（ARDS） was explored. Central vein pressure（CVP） and pulse indicator continuous cardiac output（Pi CCO） were monitored in 12 anesthetized and intubated healthy piglets. Pulmonary ARDS（ARDSp） and extrapulmonary ARDS（ARDSexp） models were respectively established by lung lavage of saline solution and intravenous injection of oleic acid. Then the piglets received HFOV for 4 h. EVLW index（EVLWI）, EVLW/intratroracic blood volume（ITBV） and pulmonary vascular permeability index（PVPI） were measured before and after modeling（T0 and T1）, and T2（1 h）, T3（2 h）, T4（3 h） and T5（4 h） after HFOV. CC16 and s ICAM-1 were also detected at T1 and T5. Results showed at T1, T3, T4 and T5, EVLWI was increased more significantly in ARDSp group than in ARDSexp group（P〈0.05）. The EVLWI in ARDSp group was increased at T1（P=0.008）, and sustained continuously within 2 h（P=0.679, P=0.216）, but decreased at T4（P=0.007） and T5（P=0.037）. The EVLWI in ARDSexp group was also increased at T1（P=0.003）, but significantly decreased at T3（P=0.002） and T4（P=0.019）. PVPI was increased after modeling in both two groups（P=0.004, P=0.012）, but there was no significant change within 4 h（T5） under HFOV in ARDSp group, while PVPI showed the increasing trends at first, then decreased in ARDSexp group after HFOV. The changes of EVLW/ITBV were similar to those of PVPI. No significant differences were found in ΔEVLWI（P=0.13）, ΔPVPI（P=0.28） and ΔEVLW/ITBV between the two groups（P=0.63）. The significant decreases in both CC16 and s ICAM-1 were found in both two groups 4 h after HFOV, but there was no significant difference between the two groups. It was concluded that EVLWI and lung capillary permeability were markedly increased in ARDSp and ARDSexp groups. EVLW could be decreased 4 h after the HFOV treatment. HFOV, EVLW/ITBV and PVPI were increased slightly at first, and then decreased in ARDSexp group, while in ARDSp group no significant difference was found after modeling. No significant differences were found in the decreases in EVLW and lung capillary permeability 4 h after HFOV.

Study on Characteristics of 3-D Translating-Pulsating Source Green Function of Deep-Water Havelock Form and Its Fast Integration Method

The singularities, oscillatory performances and the contributing factors to the 3-＇D translating-pulsating source Green function of deep-water Havelock form which consists of a local disturbance part and a far-field wave-like part, are analyzed systematically. Relative numerical integral methods about the two parts are presented in this paper. An improved method based on LOBATTO rule is used to eliminate singularities caused respectively by infinite discontinuity and jump discontinuous node from the local disturbance part function, which makes the improvement of calculation efficiency and accuracy possible. And variable substitution is applied to remove the singularity existing at the end of the integral interval of the far-field wave-like part function. Two auxiliary techniques such as valid interval calculation and local refinement of integral steps technique in narrow zones near false singularities are applied so as to avoid unnecessary integration of invalid interval and improve integral accordance. Numerical test results have proved the efficiency and accuracy in these integral methods that thus can be applied to calculate hydrodynamic performance of floating structures moving in waves.

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms,focusing on the case of multiple,non-commensurate frequencies.We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question.Numerical examples illustrate the effectiveness of the method.

Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number.Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.Simulated examples are given to illustrate the conclusion.

Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures

In this paper we analyze a long standing problem of the appearance of spurious,non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures.The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem.We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coeffi-cients to small k components would lead to the appearance of non-physical solutions.We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution.This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures.Finally,based on the above requirement of small k,we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.