Time-splitting Technique Research Articles

Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity

In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in H1-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., τ≤h2 and τ≥h2, where τ denotes the temporal size and h is the spatial size. Subsequently, the global superconvergence error estimate in H1-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

A non-hydrostatic numerical model for simulating regular wave breaking and surf-swash zone motions

In the present study, a non-hydrostatic two-dimensional vertical model has been developed to simulate the breaking of regular waves and surf-swash zone motions on a sloping beach. The objective of the present study was to estimate parameters at depth. The governing equations, based on the pressure-correction projection method, were solved in two main phases. In the initial phase, intermediate velocities were acquired through the resolution of advection–diffusion and explicit dynamic pressure gradient terms within the momentum equations, employing a time-splitting technique. To ensure local momentum conservation and solution monotonicity, modifications were made to the governing equations and the solution approach. In the second phase, through the substitution of intermediate velocities and the corrected pressure gradient term from the momentum equations into the continuity equation, with the elimination of velocities, a Poisson pressure-correction equation was derived. In the discretization stage, an innovation was proposed to compute horizontal velocities at the locations where vertical velocities are present, significantly reducing computational costs. The equation was then converted into a system of linear equations, which was solved implicitly. Comparisons between numerical results and experiments concerning plunging and spilling breakers reveal that the developed model satisfactorily simulates the outcomes.

Evolution Function Based Reach-Avoid Verification for Time-varying Systems with Disturbances

In this work, we investigate the reach-avoid problem of a class of time-varying analytic systems with disturbances described by uncertain parameters. Firstly, by proposing the concepts of maximal and minimal reachable sets, we connect the avoidability and reachability with maximal and minimal reachable sets respectively. Then, for a given disturbance parameter, we introduce the evolution function for exactly describing the reachable set, and find a series representation of this evolution function with its Lie derivatives, which can also be regarded as a series function with respect to the uncertain parameter. Afterward, based on the partial sums of this series, over- and under-approximations of the evolution function are constructed, which can be realized by interval arithmetics with designated precision. Further, we propose sufficient conditions for avoidability and reachability and design a numerical quantifier elimination based algorithm to verify these conditions; moreover, we improve the algorithm with a time-splitting technique. We implement the algorithms and use some benchmarks with comparisons to show that our methodology is both efficient and promising. Finally, we additionally extend our methodology to deal with systems with complex initial sets and time-dependent switchings. The performance of our extended method for these systems is also shown by four examples with comparisons and discussions.

Non-Darcy nanoliquid buoyant flow and entropy generation analysis in an inclined porous annulus: Effect of source-sink arrangement

In several engineering and industrial applications, the primary challenge in the design of thermal devices is to minimize the entropy production with maximum energy dissipation so that the system efficiency could be enhanced. In particular, an inclined partially heated annulus is far challenging due to the presence of both tangential and normal gravity components, which produces major impacts on the fluid movement and associated transport rates as well as entropy production. From several important applications point of view, the current investigation is mainly focused to address the combined impacts of source-sink arrangements as well as the annular inclination on buoyant nanoliquid motion, thermal dissipation along with irreversibility distribution in a tilted porous annulus. The governing model equations are numerically solved by utilizing the finite difference based time-splitting and relaxation techniques. Using second-order finite difference approximation, the partial derivatives are approximated and the resulting system of algebraic equations are inverted using Thomas algorithm. The influence of various controlling parameters such as geometric inclination angle, nanoparticle shape, Darcy number, length and position of source-sink on fluid movement, thermal and entropy distribution are scrutinized in detail. Through a wide range of numerical experiments, it has been found that the annular inclination angle and location of source-sink arrangement plays a vital role on fluidity and flow patterns of nanoliquid in the porous saturated annulus. For smaller size of source-sink pair, higher thermal dissipation with minimal entropy production could be achieved, which is an important observation for the design of thermally efficient systems. We have also noticed that a better thermal transport rate with minimum irreversibility rate could be attained for a particular source-sink arrangement, in particular for γ=0∘, the staggered positioning of source-sink pair is beneficial rather than in-line arrangement. Among the various parameters arising in this analysis, the nanoparticle shape has been one of the key parameters to test the thermal efficiency. It has been predicted that the blade shaped nanoparticle induces better thermal efficiency, among the five different shapes of nanoparticle. The outcomes of this analysis could be utilized in applications, such as solar systems, electronic equipment cooling and heat exchangers.

A Meshfree Time-Splitting Approach for the Time-Fractional Burgers’ Equation

We consider to represent an algorithm for time-fractional Burgers’ equation utilizing the multiquadric-radial basis functions with the time-splitting technique. This algorithm is performed on the three examples. The numerical results indicated that the algorithm yields accurate approximate solutions of the time-fractional Burgers’ equation. The physical behavior of the obtained solutions is given with graphics, and the results showed that the obtained solutions are in good match with the solutions reported in the literature. The algorithm is accurate, flexible, and easy to implement.

Numerical solution for Benjamin-Bona-Mahony-Burgers equation with Strang time-splitting technique

In the present manuscript, the Benjamin-Bona-Mahony-Burgers (BBMB) equation will be handled numerically by Strang time-splitting technique. While applying this technique, collocation method based on quintic B-spline basis functions is applied. In line with our purpose, after splitting the BBM-Burgers equation given with appropriate initial boundary conditions into two subequations containing the derivative in terms of time, the quintic B-spline based collocation finite element method (FEM) for spatial discretization and the suitable finite difference approaches for time discretization is applied to each subequation and hereby two different systems of algebraic equations are obtained. Four test problems are utilized to test the efficiency and reliability of the presented method. The error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$,$I_2$ and $I_3$, respectively, are computed. To do a comparison with the other studies in the literature, the newly found approximate solutions are exhibited in both tabular and graphical formats. Also, stability analysis of numerical approach by the von Neumann method is researched.

Beyond backpropagate through time: Efficient model‐based training through time‐splitting

Model-based policy gradient (MBPG) has been employed to seek an approximate solution to the optimal control problem. However, there is coupling between adjacent states due to temporal dependencies, making the training time grow linearly with the time horizon. This paper reshapes the training process of MBPG with the time-splitting technique to establish a time-independent algorithm called Training Through Time-Splitting (T3S). First, copy the coupled variables to obtain two independent variables. Meanwhile, an extra variable together with an equivalence constraint is introduced for problem consistency. Then, the transformed problem divides into subproblems with carefully derived loss functions. Subproblems own decoupled variables and shared policy networks, which means they can be optimized concurrently. Guided by the algorithm design, this paper further proposes an asynchronous parallel training scheme to accelerate training efficiency. Numerical simulation shows that the T3S algorithm outperforms the MBPG algorithm by 83.6% in wall-clock time with a trajectory tracking task.

Accelerated convergence of time‐splitting algorithm by relaxation method

The alternating direction method of multipliers (ADMM) is a widely used model predictive control (MPC) acceleration method. It adopts the time-splitting technique, splitting the original problem into independent subproblems. Relaxed ADMM (R-ADMM) is a generalization of ADMM that often achieves faster convergence. However, its parameters must be chosen by an expert user. Besides, the existing convergence proof of R-ADMM adopts a first-order Taylor approximation, which makes the range of relaxation factors conservative. We tackle these weaknesses by giving rigorous evidence and finding the optimal relaxation factor. Firstly, we deduce the convergence of the R-ADMM algorithm, yielding an accurate range of relaxation factors. Then, we analyze the relationship between convergence rate and relaxation factor and conclude that the optimal relaxation factor depends on a recurrence condition. Since splitting introduces the equality constraint, the decoupled states are getting close in the iteration, meeting the recursive requirements and helping find the optimal relaxation factor. Finally, various trajectory tracking tasks are conducted to verify the efficiency of the R-ADMM algorithm. And the simulation results show that the R-ADMM algorithm reduces the number of iterations by 63.7% compared with the ADMM algorithm in the double lane change task.

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

<p style='text-indent:20px;'>In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.</p>

ASUCA: The JMA Operational Non-hydrostatic Model

The non-hydrostatic numerical weather prediction (NWP) model ASUCA developed by the Japan Meteorological Agency (JMA) was launched into operation as 2 and 5 km-resolution regional models in 2015 and 2017, respectively. This paper outlines specifications of ASUCA with focus on the dynamical core and its configuration/accuracy as an operational model. ASUCA is designed for high computational stability and efficiency, mass conservation and forecast accuracy. High computational stability is achieved via a time-split integration scheme to compute acoustic terms and an advection scheme with a flux-limiter function to avoid numerical oscillation. In addition, vertical advection and sedimentation are calculated together with another exclusive time-splitting technique. ASUCA adopts hybrid parallelization using Message Passing Interface (MPI) and Open Multi Processing (OpenMP) for high computational efficiency on massive parallel scalar computers. The three-dimensional arrays are allocated such that the vertical direction is the stride-one innermost dimension to make effective use of cache and multi-thread parallelization. This is particularly advantageous for physical processes evaluated in a vertical column. To ensure mass conservation, density rather than pressure is integrated as a prognostic variable in flux-form fully compressible governing equations. ASUCA exhibited better performance than the previous operational model in idealized and NWP tests.

Numerical dispersion of solute transport in an integrated surface–subsurface hydrological model

Integrated surface–subsurface hydrological models (ISSHMs) are increasingly being used for the assessment of contaminant transport in the environment, in addition to their more common use in water flow applications. However, the subsurface solute transport solvers in these models are prone to numerical dispersion errors. Numerical dispersion is a well-known issue in groundwater modeling, but its impacts on the results of ISSHM simulations are still poorly understood. In this study, the CATchment HYdrology (CATHY) model is used to assess the potential impacts of numerical dispersion on the simulation of coupled surface–subsurface solute transport. We first simulate the subsurface transport of a nonreactive tracer in two soil column test cases (1D and 3D) with known analytical solutions. The subsurface solute transport solver in CATHY adopts a computationally efficient time-splitting technique whereby the advection component of the governing equation is solved on elements and the hydrodynamic dispersion component is solved on nodes. Comparison between simulation results and analytical solutions with different mesh discretizations and different values for the hydrodynamic dispersion coefficients allows for accurate quantification of the numerical dispersion error and yields insights into the parameters and other factors that control it. It is shown that, taken alone, the advection and dispersion solvers are very robust, but their combination can result in significant numerical dispersion, stemming from the exchange of concentration information from elements to nodes and vice versa in the time-splitting procedure. The tests also show that these errors can be kept under control by ensuring that the grid Péclet number is in the range 0.5-1.0 or smaller. We then apply CATHY in a third test case involving two synthetic hillslopes (concave and convex) in fully coupled surface–subsurface mode, in order to examine the impact of this subsurface numerical dispersion on simulated streamflow hydrographs, in particular with reference to pre-event water contributions to runoff. Here as well the results show that the effect of numerical dispersion can be controlled by keeping the grid Péclet number sufficiently small. This work provides a new set of benchmark test cases for integrated surface–subsurface hydrological models, extending to solute transport the flow-only suite of benchmarks recently published in two intercomparison studies.

Two-hop cognitive DF relaying with wireless power transfer in time and power domains

In this paper, we consider a wireless powered cognitive relaying system with a secondary relay (SR) capable of harvesting wireless energy. Along with an access point (AP) continuously transmitting the primary data to a primary user (PU), a secondary source (SS) can transmit the secondary data to a secondary destination (SD) with the help of SR using the decode-and-forward (DF) protocol. SR can harvest energy from both SS and AP in both time and power domains using time-splitting and power-splitting techniques. The interference from primary data transmissions can help boost the amount of harvested energy at SR. The transmit power of SS is regulated by the interference threshold at PU and the allowable peak power. Despite the above two constraints, the transmit power of SR is further constrained by the amount of harvested energy. Once SR successfully decodes the data from SS, it will forward the data to SD using a constrained power. We analyze the approximate outage probabilities for both primary and secondary systems. Simulation results are provided to verify the effectiveness of our theoretical analysis and reveal the impacts of various parameters to the outage performance.

Strang time-splitting technique for the generalised Rosenau–RLW equation

Strang time-splitting technique for the generalised Rosenau–RLW equation

Two efficient numerical methods for solving Rosenau-KdV-RLW equation

In this study, two efficient numerical schemes based on B-spline finite element method (FEM) and time-splitting methods for solving Rosenau-KdV-RLW equation are presented. In the first method, the equation is solved by cubic B-spline Galerkin FEM. For the second method, after splitting Rosenau-KdV-RLW equation in time, it is solved by Strang timesplitting technique using cubic B-spline Galerkin FEM. The differential equation system in the methods is solved by the fourth-order Runge-Kutta method. The stability analysis of the methods is performed. Both methods are applied to an example. The obtained numerical results are compared with some methods available in the literature via the error norms and , convergence rates, and mass and energy conservation constants. The present results are found to be consistent with the compared ones.

A time-splitting pressure-correction projection method for complete two-fluid modeling of a local scour hole

A two-dimensional vertical (2DV), Eulerian two-phase model or complete two-fluid model of the free surface flow was developed to simulate water-sediment flow in a local scour hole. In the model, the complete forms of the vertical, two-dimensional, two-fluid Navier-Stokes equations were discretized using a finite volume scheme. This discretization was done based on a standard staggered grid system using a curvilinear network system in compliance with the bed boundaries and water level. At the beginning of the computational cycle, the equations governing the fluid phase were solved based on the two-step projection method with a pressure-correction technique. In the first step, the intermediate fluid velocities were obtained by solving different phases of the momentum equations of the fluid phase using the time-splitting technique. In the second step, pressure was obtained and fluid velocities were updated. In this step a simple discretization method was applied for decreasing the computational complexity. After obtaining all the fluid phase variables at a new time step, the sediment phase momentum equations were solved using the time-splitting technique and sediment velocities were obtained. Then, at the end of the computational cycle, the sediment phase mass equation was solved and the concentrations of both phases were updated. At last, the capacity of the model for simulating of the longitudinal fluid velocity and sediment concentration in a local scour hole was evaluated. Numerical results were found to be in good agreement with experimental data.

An accurate and efficient numerical method for solving linear peridynamic models

• An accurate and efficient numerical method for linear peridynamic models. • Coupling of time-splitting technique and localized radial basis function method. • Valid for any dimension, irregular domain, large time step, and long execution time. • More accurate, efficient and robust than existing numerical methods. • Performance of the method demonstrated with some representative examples. In this paper, we combine the recently developed localized radial basis functions-based pseudo-spectral method with the time-splitting technique to solve a linear wave equation arising from modelling the wave dynamics using peridynamic formulation in continuum mechanics . Specifically, we adopt this combined method for solving a Hamiltonian ordinary differential equation system, which is equivalent to the original linear peridynamic equation after introducing a new variable. The proposed approach inherits advantages of these two related methods in space and time: (1) offering high accuracy and efficiency in the solution of the problem under irregular domains for both uniform and non-uniform discretizations ; (2) extending the applicability of the approach to multi-dimensions; and (3) maintaining a good approximation for problems at large time-step and long time integration. Numerical results indicate that the proposed method is simple, accurate, efficient, and stable for solving various linear peridynamic problems.

Numerical Simulation and Error Estimation of the Time-Dependent Allen–Cahn Equation on Surfaces with Radial Basis Functions

In this paper a numerical simulation based on radial basis functions is presented for the time-dependent Allen–Cahn equation on surfaces with no boundary. In order to approximate the temporal variable, a first-order time splitting technique is applied. The error analysis is given when the true solution lies on appropriate Sobolev spaces defined on surfaces. The method only requires a set of scattered points on a given surface and an approximation to the surface normal vectors at these points. Besides, the approach is based on Cartesian coordinates and thus any coordinate singularity has been omitted. Some numerical results are given to illustrate the ability of the technique on sphere, torus and red blood cell as three well-known surfaces.

Analytical and field verification of a 3D hydrodynamic and water quality numerical scheme based on the 2D formulation in CE-QUAL-W2

ABSTRACTA new 3D hydrodynamic model was developed to simulate water quality transport in surface waterbodies. The governing equations are the continuity equation, free surface equation, momentum equation and transport equation. The 2D numerical scheme of CE-QUAL-W2 was expanded in three dimensions and modified to solve for the free surface elevation. A time splitting technique was employed to solve the momentum and transport equation. The numerical formulation of the 3D scheme used a novel solution, which resulted in a tri-diagonal matrix form for solving the free surface equation rather than a more computationally intensive penta-diagonal matrix solution. In addition, the hydrodynamic and water quality equations were solved at the same time step in order to allow feedback between water quality and hydrodynamics. The verification of the model hydrodynamics and temperature was performed by comparing the model predictions to known analytical solutions and field data from Lake Chaplain, Washington, USA. There was good agreement of the solution of the hydrodynamic equations to analytical solutions and field data.

An algorithm for partial functional differential equations modeling tumor growth

We introduce a parallel algorithm for the numerical simulation of the growth of human tumor cells in time-varying environments and their response to therapy. The behavior of the cell populations is described by a system of delay partial differential equations with time-dependent coefficients. We construct the new algorithm by developing a time-splitting technique in which the entire problem is split into independent tasks assigned to arbitrary numbers of processors chosen in light of available resources. We present the results of a series of numerical experiments, which confirm the efficiency of the algorithm and exhibit a substantial decrease in computational time thus providing an effective means for fast clinical, case-by-case applications of tumor invasion simulations and possible treatment.