Order Operator Splitting Research Articles

WAVEx: Stabilized finite elements for spectral wind wave models using FEniCSx

The prediction of the wind wave spectrum of the ocean using numerical models are an important tool for researchers, engineers, and communities living in coastal areas. The governing equation of the wind wave models, the Wave Action Balance Equation, presents unique challenges for implementing reliable numerical models because it is highly advective, highly nonlinear and high dimensional. Historically, most operational models have utilized finite difference methods, others have used finite volume methods but relatively few attempts at using finite element methods. In this work, we seek to fill this gap by investigating several different finite element discretizations of the Wave Action Balance Equation. The methods, which include streamline upwind Petrov–Galerkin (SUPG), least squares, and discontinuous Galerkin, are implemented and convergence properties are examined for some simplified 2-D test cases. Then, a new spectral wind wave model, WAVEx, is formulated and implemented for the full problem setting. WAVEx uses continuous finite elements along with SUPG stabilization in geographic/spectral space that allows for fully unstructured triangular meshes in both geographic and spectral space. For propagation in time, a second order fully implicit finite difference method is used. When source terms are active, a second order operator splitting scheme is used to linearize the problem. In the splitting scheme, propagation is solved using the implicit method and the nonlinear source terms are treated explicitly. Several test cases, including analytic tests and laboratory experiments, are demonstrated and results are compared to analytic solutions, observations, as well as output from another model that is used operationally.

A high order operator splitting method based on spectral deferred correction for the nonlocal viscous Cahn-Hilliard equation

Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method.

Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Operator Splitting Around Euler-Maruyama Scheme and High Order Discretization of Heat Kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

A second order operator splitting numerical scheme for the “good” Boussinesq equation

The nonlinear stability and convergence analyses are presented for a second order operator splitting scheme applied to the “good” Boussinesq equation, coupled with the Fourier pseudo-spectral approximation in space. Due to the wave equation nature of the model, we have to rewrite it as a system of two equations, for the original variable u and v=ut, respectively. In turn, the second order operator splitting method could be efficiently designed. A careful Taylor expansion indicates the second order truncation error of such a splitting approximation, and a linearized stability analysis for the numerical error function yields the desired convergence estimate in the energy norm. In more details, the convergence in the energy norm leads to an ℓ∞(0,T⁎;H2) convergence for the numerical solution u and ℓ∞(0,T⁎;ℓ2) convergence for v=ut. And also, the presented convergence is unconditional for the time step in terms of the spatial grid size, in comparison with a severe time step restriction, Δt≤Ch2, required in many existing works.

Higher order operator splitting methods for an image de-noising model

This paper is concerned with fast iterative methods with development of Euler-Lagrange equation which results from the minimization of Rudin-Osher-Fatemi (ROF) model. There are many applications of image de-noising in field of medical and astronomy. We can classify the image de-noising models into additive and multiplicative noise removal models. In case of additive noise, we have an image u corrupted with additive Gaussian noise η, the main task is to recover u from the image formation model u0 = u+η. This paper mainly focus on additive noise removal. Here semi-implicit (SIM), additive operator splitting (AOS) and additive multiplicative operator splitting (AMOS) type schemes are developed. The quality in AOS is, it treats with all coordinate axes in an equal manner. We develop a new AMOS scheme for the solution of Euler-Lagrange equation arisen from minimization of image additive noise removal model. Comparison of AMOS with SIM and AOS is also presented. Experimental results shows that by using AMOS, additive noisy image can be de-noised with best results. Numerical examples are given to show gain in CPU timing and fast convergence of AMOS-based algorithm. https://doi.org/10.28919/jmcs/3312

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville–von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born–Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics ‘on the fly’, including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found. Program summaryProgram Title: WavePacketProgram Files doi:http://dx.doi.org/10.17632/tc5mrhx9sb.1Licensing provisions: GPLv3Programming language:MatlabNature of problem:Schrödingerś equations are of fundamental importance in non-relativistic quantum mechanics of distinguishable particles. The solutions of the time-independent equation (TISE) are wavefunctions in coordinate space, the absolute squares of which are usually interpreted as probability density. The time-dependent equation (TDSE) describes the dynamics of a quantum system evolving in time. It plays a crucial role for the simulation, understanding, and prediction of modern experiments in atomic, molecular and optical physics where systems are driven by temporally shaped external fields.Solution method:All numerical methods compiled in WavePacket are based on discrete variable representations. Currently implemented are Gauss–Hermite, Gauss–Legendre and FFT-based schemes. The TISE is solved either by direct diagonalization or by propagation in imaginary time. For the TDSE there is a choice between second order differencing, operator splitting and Chebychev polynomial methods.Additional comments including Restrictions and Unusual features:The WavePacket program package is rather easy and intuitive to use, providing visualization of quantum dynamics ‘on the fly’. It is mainly intended for low-dimensional systems, typically not exceeding three to five degrees of freedom. Detailed user guides and reference manuals are available through numerous Wiki pages hosted at the SourceForge platform where also a large number of well documented demonstration examples can be found.

First and second order operator splitting methods for the phase field crystal equation

In this paper, we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two-phase systems on atomic length and diffusive time scales. A core idea of the methods is to decompose the original equation into linear and nonlinear subequations, in which the linear subequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used. By combining these subequations, we achieve the first- and second-order accuracy in time. We present numerical experiments to show the accuracy and efficiency of the proposed methods.

High order operator splitting methods based on an integral deferred correction framework

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by r with each correction, where r is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial–boundary value problems are presented to demonstrate the performance of the proposed IDC approach.

A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms

Allen–Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decompose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.

Simulation of Transrib HIFU Propagation and the Strategy of Phased-array Activation

Liver ablation is challenging in high-intensity focused ultrasound (HIFU) because of the presence of ribs and great inhomogeneity in multi-layer tissue. In this study, angular spectrum approach (ASA) has been used in the wave propagation from phased-array HIFU transducer, and diffraction, attenuation and the nonlinearity are accounted for by means of second order operator splitting method. Bioheat equation is used to simulate the subsequent temperature elevation and lesion formation with the formation of shifted focus and multiple foci. In summary, our approach could simulate the performance of phased-array HIFU in the clinics and then develop an appropriate treatment plan.

Simulations of focused shear shock waves in soft solids and the brain

Because of a very small speed, shear waves in soft solids are extremely nonlinear, with nonlinearities four orders of magnitude larger than in classical solids. Consequently, these nonlinear shear waves can transition from a smooth to a shock profile in less than one wavelength. We hypothesize that traumatic brain injuries (TBI) could be caused by the sharp gradients resulting from shear shock waves. However, shear shock waves are not currently modeled by simulations of TBI. The objective of this paper is to describe shear shock wave propagation in soft solids within the brain, with source geometry determined by the skull. A 2D nonlinear paraxial equation with cubic nonlinearities is used as a starting point. We present a numerical scheme based on a second order operator splitting which allows the application of optimized numerical methods for each terms. We then validate the scheme with Guiraud's nonlinear self-similarity law applied to cusped caustics. Once validated, the numerical scheme is then applied to a blast wave problem. A CT measurement of the human skull is used to determine the initial conditions and shear shock wave simulations are presented to demonstrate the focusing effects of the skull geometry.

Local error estimates for adaptive simulation of the reaction–diffusion master equation via operator splitting

The efficiency of exact simulation methods for the reaction–diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.

Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

Applications of a higher order operator splitting scheme on parabolic-equation methods for modeling underwater sound propagation

A higher order operator splitting scheme is presented to increase the accuracy of parabolic-equation (PE) methods. This splitting scheme is essentially applied to the square-root Helmholtz operator resulting from the PE approximation, and it can benefit both the split-step Fourier and split-step Padé methods for modeling sound propagation in three-dimensional environments. Note that the square-root Helmholtz operator in the split-step Fourier PE method is split into a free propagator and a phase anomaly, but in the split-step Padé PE method it is split into two one-dimensional transverse derivatives. Our higher order scheme can handle these two different types of operator splittings, and accuracy improvements come from the fact that our computation includes (to second order) cross terms of the split operators. Numerical underwater acoustic examples are provided to demonstrate the performance of this scheme and comparisons against other schemes. [Work supported by the Office of Naval Research.]

Numerical Investigation of Superharmonic Imaging Using Chirp Excitation

At present, along with the fundamental method of ultrasound imaging, tissue harmonic imaging based on imprinting of the second harmonic component is established as the standard method of ultrasound imaging. Recently introduced superharmonic imaging (SHI) has an edge over this standard method and has improved spatial resolution. To explore further advancement in ultrasound imaging for enhanced resolution and penetration depth. SHI is the summation of higher harmonics generated due to nonlinear tissue propagation of ultrasound. Here, the addition of third, fourth and fifth harmonics has been done for the superharmonic field. The modeling of nonlinear wave propagation is done in frequency domain on the basis of the second order operator splitting approach. The higher harmonic components are picked up using the inverse filters. The standard evaluation parameters were significantly improved with linear frequency modulation (FM) chirp. The axial level was 2.8 times higher with linear FM chirp as compared to conventional short pulse. Minor benefits such as a 5% increase in beam width and suppression of side lobes were also recorded. This numerical investigation confirms that linear FM chirp in SHI provides better resolution and penetration depth. State of the art fundamental and second harmonic ultrasound imaging techniques were also improved with linear FM chirp as compared with conventional excitation. SHI is superior to the existing diagnostic ultrasound techniques.

Higher order operator splitting methods via Zassenhaus product formula: Theory and applications

In this paper, we contribute higher order operator splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative splitting methods. The underlying analysis to obtain higher order operator splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods. The verification of the improved splitting methods are done with numerical examples. An individual handling of each operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order operator splitting methods.

An application of some high order operator splitting methods to systems of stiff differential equations

AbstractIn this work we compare operator splitting methods of high order that are applied to problems with stiff matrices. In order to efficiently solve the resultant subproblems is possible to use implicit Runge‐Kutta methods. We apply an alternative extrapolation technique that works well for the tested problems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fully Discrete Finite Element Approximation for Anisotropic Surface Diffusion of Graphs

We analyze a fully discrete numerical scheme for approximating the evolution of graphs for surfaces evolving by anisotropic surface diffusion. The scheme is based on the idea of second order operator splitting for the nonlinear geometric fourth order equation. This yields two coupled spatially second order problems, which are approximated by linear finite elements. The time discretization is semi-implicit. We prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis and illustrate surface diffusion.

Fourth order operator splitting method for the three space parabolic equation with variable coefficients

This paper deals with two new compact difference methods of 0(k 2 + kh 2 + h 4) for solving the system of general three space non-linear parabolic equations subject to the appropriate initial and Dirichlet boundary conditions using 27 and 19 spatial grid points in the presence and absence of mixed derivative terms, respectively. We also discuss the fourth order operator splitting method for the linear equation when applied to the proposed difference method. Various numerical examples including 3-space unsteady Navier-Stokes equations are provided which demonstrate the effectiveness and the computational efficiency of the methods developed.