Operator Splitting Scheme Research Articles

An operator splitting scheme for numerical simulation of spinodal decomposition and microstructure evolution of binary alloys

This article compares the operator splitting scheme to linearly stabilized splitting and semi-implicit Euler's schemes for the numerical solution of the Cahn-Hilliard equation. For the purpose of validation, the spinodal decomposition phenomena have been simulated. The efficacy of the three schemes has been demonstrated through numerical experiments. The computed results show that the schemes are conditionally stable. It has been observed that the operator splitting scheme is computationally more efficient.

QPC-TDSE: A parallel TDSE solver for atoms and small molecules in strong lasers

The QPC-TDSE program serves as a general tool to study laser-driven dynamics of electrons in ideal isolated atoms and molecules by solving the full-dimensional non-relativistic time-dependent Schrödinger equation (TDSE) within single-active-electron approximation. It expands the full-dimensional electronic wavefunction in spherical coordinates by spherical harmonics and B-spline functions and employs a set of parallel Crank-Nicolson propagators combined with split-operator techniques to evolve the wavefunction in time, which support centrifugal and multi-polar static potentials to treat atomic and molecular scenarios and accepts arbitrary combinations of linearly or elliptically polarized lasers within the dipole approximation. The program is capable of extracting the photo-electron momentum distribution via t-SURFF approach or projection onto either the exact scattering states or the planewaves. Its applications in different scenarios are given as examples, e.g., above threshold ionization, attosecond clock, higher-order harmonic generation. Program summaryProgram Title: QPC-TDSECPC Library link to program files:https://doi.org/10.17632/xjm3kfgv75.1Licensing provisions: GPLv3Programming language: C++External libraries: HDF5, GSL, MKLNature of problem: Numerical solution of TDSE and extraction of various types of electron spectrum.Solution method: The electronic wavefunction is expanded by B-spline functions and spherical harmonics whose range is chosen elaborately to reduce the total number of partial waves for non-linearly polarized lasers. The Crank-Nicolson approach combined with an operator-splitting scheme is used to propagate the wavefunction in time, either in velocity gauge or length gauge. Matrix inversions are solved via either dense or sparse linear algebra solvers according to their structures. The t-SURFF method and projections onto either the scattering states or planewaves are provided for the accurate extraction of the momentum distributions.Additional comments including restrictions and unusual features: Only lasers within dipole approximation are supported. For the multi-polar potentials, only pure Coulombic ones are supported. Routines for solving exact scattering states have only been implemented for centrifugal potentials. The codes are written in C++17 and can only be compiled on the platforms that support the avx instruction sets. An extension for the propagation algorithm using the avx-512 intrinsics is provided as optional.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

Convergence analysis of the splitting method to the nonlinear heat equation

AbstractIn this article, we analyze an operator splitting scheme of the nonlinear heat equation in (): where and with and . We establish the well‐posedness of the approximation of in ‐space (), and furthermore, we derive its convergence rate of order for a time step . Finally, we give some numerical examples to confirm the reliability of the analyzed result

The operator splitting schemes revisited: primal-dual gap and degeneracy reduction by a unified analysis

The operator splitting schemes revisited: primal-dual gap and degeneracy reduction by a unified analysis

A simple and efficient adaptive time stepping technique for low-order operator splitting schemes applied to cardiac electrophysiology.

We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude-found in a previous one-dimensional analytical study by Spiteri and Ziaratgahi-to construct the new time step controller for three-dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional-integral-differential controllers using state-of-the-art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.

A numerical efficient splitting method for the solution of HIV time periodic reaction–diffusion model having spatial heterogeneity

This study examines a novel reaction–diffusion model for the existence and treatment of acquired immunodeficiency syndrome. This model is a spatial extension of the recent HIV model and human immunodeficiency viruses cause this disorder. The most significant barrier to eradicating this virus is latency and the virus’ subsequent viral reservoir in CD4+ T cells. A nonstandard operator splitting strategy is proposed to approximate the solution of the time-periodic reaction–diffusion model. The main advantages of employing this approach over other techniques are its low computational costs, high accuracy and ease of implementation. The results are truly solid and match those available in the literature. The nature of the solution for the threshold parameter is demonstrated graphically using numerical results. Finally, the M-matrix theory and the positivity of the proposed scheme are discussed.

An effective operator splitting method based on spectral deferred correction for the fractional Gray–Scott model

This paper presents a method by combining the semi-implicit spectral deferred correction (SDC) method with the operator splitting scheme to simulate the fractional Gray-Scott (GS) model. We start with the second-order operator splitting scheme, which is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via the Crank-Nicolson formula and Rubin-Graves linearization technique, which can be solved effectively. The stability and convergence of this method are analyzed in L2-norm. Moreover, the scheme also takes advantage of the semi-implicit SDC method to improve the temporal accuracy. Numerical results are given to illustrate that the proposed method is a practical, accurate and efficient simulation tool for solving fractional GS problems.

Elastic net-based non-negative iterative three-operator splitting strategy for Cerenkov luminescence tomography.

Cerenkov luminescence tomography (CLT) provides a powerful optical molecular imaging technique for non-invasive detection and visualization of radiopharmaceuticals in living objects. However, the severe photon scattering effect causes ill-posedness of the inverse problem, and the location accuracy and shape recovery of CLT reconstruction results are unsatisfactory for clinical application. Here, to improve the reconstruction spatial location accuracy and shape recovery ability, a non-negative iterative three operator splitting (NNITOS) strategy based on elastic net (EN) regularization was proposed. NNITOS formalizes the CLT reconstruction as a non-convex optimization problem and splits it into three operators, the least square, L1/2-norm regularization, and adaptive grouping manifold learning, then iteratively solved them. After stepwise iterations, the result of NNITOS converged progressively. Meanwhile, to speed up the convergence and ensure the sparsity of the solution, shrinking the region of interest was utilized in this strategy. To verify the effectiveness of the method, numerical simulations and in vivo experiments were performed. The result of these experiments demonstrated that, compared to several methods, NNITOS can achieve superior performance in terms of location accuracy, shape recovery capability, and robustness. We hope this work can accelerate the clinical application of CLT in the future.

A Second-Order Accurate, Operator Splitting Scheme for Reaction-Diffusion Systems in an Energetic Variational Formulation

A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting scheme is proposed and analyzed for reaction-diffusion systems with the detailed balance condition. The scheme is designed based on an energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. At the reaction stage, the reaction trajectory equation is approximated by a second-order Crank--Nicolson type method. The unique solvability, positivity-preserving property, and energy stability are established based on a convexity analysis. In the diffusion stage, an exact integrator is applied if the diffusion coefficients are constant, and a Crank--Nicolson type scheme is constructed if the diffusion part is nonlinear. In either case, both the positivity-preserving property and energy stability could be theoretically established. Moreover, a combination of the numerical algorithms at both stages by the Strang splitting approach leads to a second-order accurate, structure-preserving scheme for the original reaction-diffusion system. Numerical experiments are presented, which demonstrate the accuracy and the energy stability of the proposed scheme.

Some extensions of the operator splitting schemes based on Lagrangian and primal–dual: a unified proximal point analysis

By revisiting some popular operator splitting ideas, we present several classes of splitting schemes based on the Lagrangian, primal–dual and hybrid formulations, from which can be recovered many existing algorithms, including alternating direction method of multipliers and primal–dual hybrid gradient algorithms. In particular, we show that the generalized proximal point framework is at the root of many past and recent splitting algorithms allowing for an elementary convergence analysis of these methods through a unified scheme. The numerical tests on constrained total variation minimization show that the proposed algorithms could offer more freedom in parameter selections and, thus, achieve faster convergence speed.

Convergence Analysis of the Variational Operator Splitting Scheme for a Reaction-Diffusion System with Detailed Balance

We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu et al.,J. Comput. Phys., 436, 110253, 2021] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms, which are treated implicitly in the chemical reaction stage. In addition, a combination of a rough error estimate and a refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting results in the convergence estimate of the numerical scheme for the full reaction-diffusion system.

A dynamic relaxation method with operator splitting and random-choice strategy for SPH

In this paper, we propose an efficient dynamic relaxation method for smoothed particle hydrodynamics (SPH) to address the time-consuming issue when dealing with achieving the equilibrium of a dynamic system. First, an artificial-viscosity-based damping term is added into the momentum equation to dissipate the velocity gradient without losing the momentum conservation of the system. Two operator splitting methods, which update velocities implicitly, are then developed to discretize the added damping term for relaxing the time-step limit. To further improve the computational efficiency, a random-choice strategy is also introduced into the dynamic relaxation process, by which the damping is imposed randomly rather than each time step. Furthermore, a splitting cell-linked list scheme is proposed by dividing the conventional cell-linked list into several blocks to avoid the thread conflict when shared-memory parallelization is applied to accelerate the computation. A number of benchmark tests show the fast and efficient feature of the present method for dynamic systems achieving their equilibrium.

An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equation

An effective operator splitting scheme for solving the nonlocal Allen–Cahn equation with a Lagrange multiplier is studied. Firstly, based on the operator splitting method, the original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, respectively. Then, the nonlinear equation is solved analytically, the nonlocal equation is discretized using Crank–Nicolson format, and the Lagrange multiplier equation is solved explicitly. The mass conservation and convergence analysis of the numerical algorithm are analyzed theoretically. Numerical experiments are presented to confirm the validity of the proposed method, including the convergence, energy decline and mass conservation.

A modularized numerical framework for the process-based total system performance assessment of geological disposal systems

This study developed a safety assessment tool for geological disposal systems called APro, a systemically integrated modeling system based on modularizing and coupling the processes which need to be considered in a geological disposal system. Thermal, hydraulic, chemical, canister failure, radionuclide release and transport processes were considered in the current version of APro. Each of the unit processes in APro consists of a single Default Module, and several Alternative Modules which can increase the flexibility of the model. As an initial stage of developing the modularization concept and modeling interface, the Default Modules of each unit process were described, with one Alternative Module of chemical process. The computation part of APro is mainly a MATLAB workspace controlling COMSOL and PHREEQC, which are coupled by an operator splitting scheme. The APro model domain is a stylized geological disposal system employing the Swedish disposal concept (KBS-3 type), but the repository layout can be freely adjusted. In order to show the applicability of APro to the total system performance assessment of geological disposal system, some sample simulations were conducted. From the results, it was confirmed that coupling of the thermal and hydraulic processes and coupling of the canister failure and the radionuclide release processes were well reflected in APro. In addition, the technical connectivity between COMSOL and PHREEQC was also confirmed.

A high-order unified stochastic particle method based on the Bhatnagar-Gross-Krook model for multi-scale gas flows

Stochastic particle methods, such as Direct Simulation Monte Carlo (DSMC), are successfully used to simulate rarefied gas flows. However, since traditional stochastic particle methods employ the operator splitting scheme, i.e., decoupling the molecular motion and collisions, they are of low order accuracy in the continuum regime. In contrast, the discrete velocity method can easily construct high order schemes, but its computational and memory cost is quite high due to the huge number of discrete velocity nodes, especially for the high Mach number flows. To improve the efficiency and accuracy of the simulation for multi-scale flows, a high order stochastic particle BGK method was developed in this paper. The DIRK (Diagonally Implicit Runge–Kutta) and WLS-ENO (weighted-least-squares based essentially non-oscillatory) schemes were used to achieve a high order of accuracy in time evolution and space interpolation, respectively. However, the implementation of DIRK and WLS-ENO with computational particles is not straightforward due to the statistical nature of the stochastic particle methods. To circumvent these difficulties, firstly, the micro-macro decomposition of the relaxation term of the BGK model, which was proposed in the unified stochastic particle BGK (USP-BGK) method, was employed. The fluid limit part of the relaxation term was treated following the DIRK scheme, while the kinetic part was solved with an exact integration scheme similar to the traditional stochastic particle BGK (SP-BGK) method. Therefore, in the continuum regime, the present stochastic particle method has the same asymptotic preserving and asymptotic accurate properties as the high order semi-Lagrangian method; and in the rarefied regime, it reduces to the traditional SP-BGK method. Additionally, to overcome the effect of thermal noise in the space interpolation, a new discontinuity indicator, which considers the influence of the thermal fluctuation, was also proposed for the WLS-ENO scheme. Several 1D and 2D benchmark problems with different Mach and Knudsen numbers were tested and analyzed, such as shock and sine wave interaction, Taylor-Green vortex flow and 2D Riemann problem. Compared to the traditional SP-BGK method, this proposed method can achieve a higher order of accuracy and significantly improve the computational efficiency in the continuum regime.

Using the Split Bregman Algorithm to Solve the Self-repelling Snakes Model

Preserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

A new anisotropic fourth-order diffusion equation model based on image features for image denoising

<p style='text-indent:20px;'>Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.</p>

A numerical study of COVID-19 epidemic model with vaccination and diffusion.

The coronavirus infectious disease (or COVID-19) is a severe respiratory illness. Although the infection incidence decreased significantly, still it remains a major panic for human health and the global economy. The spatial movement of the population from one region to another remains one of the major causes of the spread of the infection. In the literature, most of the COVID-19 models have been constructed with only temporal effects. In this paper, a vaccinated spatio-temporal COVID-19 mathematical model is developed to study the impact of vaccines and other interventions on the disease dynamics in a spatially heterogeneous environment. Initially, some of the basic mathematical properties including existence, uniqueness, positivity, and boundedness of the diffusive vaccinated models are analyzed. The model equilibria and the basic reproductive number are presented. Further, based upon the uniform and non-uniform initial conditions, the spatio-temporal COVID-19 mathematical model is solved numerically using finite difference operator-splitting scheme. Furthermore, detailed simulation results are presented in order to visualize the impact of vaccination and other model key parameters with and without diffusion on the pandemic incidence. The obtained results reveal that the suggested intervention with diffusion has a significant impact on the disease dynamics and its control.

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

<p style='text-indent:20px;'>In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.</p>