An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface inste...An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiseale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.展开更多
The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high...The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high,this is a difficult multiscale problem.The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive.Fortunately,there are computationally much less costly models,that are good approximations of many wave equations precisely for very high frequencies.Even for linear wave equations these models are often nonlinear.The goal of this paper is to review such mathematical models for high frequency waves,and to survey numerical methods used in simulations.We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations.We will also discuss finite frequency corrections and some other models.展开更多
In the semiclassical regime,solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory.The number of grid points required for resolving the oscillations may become very large ev...In the semiclassical regime,solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory.The number of grid points required for resolving the oscillations may become very large even for simple model problems,making solution on a grid intractable.Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth.However,when the potential has variations on a small length-scale,quantum phenomena become important.Then asymptotic methods are less accurate.The two classes of methods perform well in different parameter regimes.This opens for hybrid methods,using Gaussian beams where we can and finite differences where we have to.We propose a new method for treating the coupling between the finite difference method and Gaussian beams.The new method reduces the needed amount of overlap regions considerably compared to previous methods,which improves the efficiency.展开更多
文摘An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiseale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.
文摘The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high,this is a difficult multiscale problem.The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive.Fortunately,there are computationally much less costly models,that are good approximations of many wave equations precisely for very high frequencies.Even for linear wave equations these models are often nonlinear.The goal of this paper is to review such mathematical models for high frequency waves,and to survey numerical methods used in simulations.We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations.We will also discuss finite frequency corrections and some other models.
文摘In the semiclassical regime,solutions to the time-dependent Schrodinger equation for molecular dynamics are highly oscillatory.The number of grid points required for resolving the oscillations may become very large even for simple model problems,making solution on a grid intractable.Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth.However,when the potential has variations on a small length-scale,quantum phenomena become important.Then asymptotic methods are less accurate.The two classes of methods perform well in different parameter regimes.This opens for hybrid methods,using Gaussian beams where we can and finite differences where we have to.We propose a new method for treating the coupling between the finite difference method and Gaussian beams.The new method reduces the needed amount of overlap regions considerably compared to previous methods,which improves the efficiency.
