Time-splitting Technique Research Articles

A numerical technique for applying time splitting methods in shallow water equations

In this paper we analyze the use of time splitting techniques for solving shallow water equations. We discuss some properties that these schemes should satisfy so that interactions between the source term and the shock waves are controlled. This work shows that these schemes must be well balanced in the meaning expressed by Greenberg and Leroux [7]. More specifically, we analyze in what cases it is enough to verify an Approximate C-property and in which cases it is required to verify an Exact C-property (see [1, 2]). We also discuss this technique in two dimensions and include some numerical tests in order to justify our argument.

Existence and approximation for Navier–Stokes system with Tresca’s friction at the boundary and heat transfer governed by Cattaneo’s law

We consider an unsteady non-isothermal incompressible fluid flow. We model heat conduction with Cattaneo’s law instead of the commonly used Fourier’s law, in order to overcome the physical paradox of infinite propagation speed. We assume that the fluid viscosity depends on the temperature, while the thermal capacity depends on the velocity field. The problem is thus described by a Navier–Stokes system coupled with the hyperbolic heat equation. Furthermore, we consider non-standard boundary conditions with Tresca’s friction law on a part of the boundary. By using a time-splitting technique, we construct a sequence of decoupled approximate problems and we prove the convergence of the corresponding approximate solutions, leading to an existence theorem for the coupled fluid flow/heat transfer problem. Finally, we present some numerical results.

Comparison of numerical methods for the Zakharov system in the subsonic limit regime

We compare numerically spatial/temporal resolution of various methods for solving the Zakharov system (ZS) in the subsonic limit regime, which involves a small parameter 0 < ε ≤ 1 inversely proportional to the acoustic speed. In this regime, i.e., 0 < ε ≪ 1 , the solution presents highly oscillatory initial layers due to the wave operator or the incompatibility of the initial data. Specifically, the solution propagates waves with wavelength of O ( ε ) and O ( 1 ) in time and space, respectively. By applying the sine pseudospectral discretization for spatial derivatives followed by a time-splitting technique for integrating the Schrödinger equation combined with an exponential wave integrator in phase space for integrating the wave equation, we propose four different numerical methods for the ZS based on different quadrature rules for approximating the integral or some property of conservation. Numerical results suggest that all the methods are spectrally accurate in space, which is uniformly for ε ∈ ( 0 , 1 ] . For temporal error, the best method converges uniformly with linear convergence rate at O ( τ ) in the subsonic limit regime.

A Novel Approach with Time-Splitting Spectral Technique for the Coupled Schrödinger–Boussinesq Equations Involving Riesz Fractional Derivative**Supported by NBHM, Mumbai, under Department of Atomic Energy, Government of India vide Grant No. 2/48(7)/2015/NBHM (R.P.)/R&D II/11403

In the present paper the Riesz fractional coupled Schrödinger–Boussinesq (S-B) equations have been solved by the time-splitting Fourier spectral (TSFS) method. This proposed technique is utilized for discretizing the Schrödinger like equation and further, a pseudospectral discretization has been employed for the Boussinesq-like equation. Apart from that an implicit finite difference approach has also been proposed to compare the results with the solutions obtained from the time-splitting technique. Furthermore, the time-splitting method is proved to be unconditionally stable. The error norms along with the graphical solutions have also been presented here.

Numerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $\tau=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $\tau=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.

An Efficient Unconditionally Stable FDTD Method for Analyzing Composite Right Left Handed Transmission Line

For analyzing the composite right/left-handed (CRLH) transmission line (TL) problems, an efficient unconditionally stable (US) finite-difference time-domain method is proposed in the letter. First, based on Kirchhoff's voltage law and Kirchhoff's current law, the CRLH TL equations are derived and rewritten in matrix form. Then, by adopting the Crank-Nicolson time split technique, and carrying out some transformations to achieve an equivalent formulation, the US scheme for the implementation of the CRLH TL equations can be obtained. Two numerical examples are used to validate the efficiency and accuracy of the proposed method in analyzing a CRLH TL and pure left-handed TL.

Numerical routing of tracer concentrations in rivers with stagnant zones

Modeling pollution transmission in rivers is an important subject in environmental engineering studies. Numerical approaches to modeling pollution transmission in rivers are useful tools for managing the water quality. The advection-dispersion equation is the governing equation in the transport of pollution in rivers. Recently, due to advances in fractional calculus in engineering modeling, the simulation of pollution transmission in rivers has been improved using the fractional derivative approach. In this study, by solving the fractional advection-dispersion equation (FRADE), a numerical model was developed for the simulation of pollution transmission in rivers with stagnant zones. To this purpose, both terms of the FRADE equation (advection and fractional dispersion) were discretized separately and the results were connected using a time-splitting technique. The analytical solution of a modified advection-dispersion equation (MADE) model and observed data from the Severn River in the UK were used to demonstrate the model capabilities. Results indicated that there is a good agreement between observed data, the analytical solution of the MADE model, and the results of the developed numerical model. The developed numerical model can accurately simulate the long-tailed dispersion processes in a natural river.

An efficient time-splitting compact finite difference method for Gross–Pitaevskii equation

We propose an efficient time-splitting compact finite difference method for Gross–Pitaevskii equation (GPE). In our method, we solve the GPE in time with time-splitting technique and in space by the compact finite difference method. To find the numerical solution of the resulting discretized system in one-dimension (1D), two-dimensions (2D) and three-dimensions (3D), we apply the fast discrete Sine transform in 1D, 2D and 3D respectively and get an efficient solver for the discretized system in 1D, 2D and 3D, respectively. Our numerical algorithm at every time step does not need linear-algebraic-equations-solver, whose computation cost will be much higher when the spatial dimension increases. The method also has the merit that it is unconditionally stable and conservative. Moreover the method can achieve spectral-like accuracy in space when high-order compact finite difference method is applied. Extensive numerical tests for the GPE in 1D, 2D and 3D are presented to demonstrate the power and accuracy of the proposed numerical method.

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.

A Multilevel Adaptive Reaction-splitting Simulation Method for Stochastic Reaction Networks

In this work, we present a novel multilevel Monte Carlo method for kinetic simulation of stochastic reaction networks characterized by having simultaneously fast and slow reaction channels. To produce efficient simulations, our method adaptively classifies the reactions channels into fast and slow channels. To this end, we first introduce a state-dependent quantity named level of activity of a reaction channel. Then, we propose a low-cost heuristic that allows us to adaptively split the set of reaction channels into two subsets characterized by either a high or a low level of activity. Based on a time-splitting technique, the increments associated with high-activity channels are simulated using the tau-leap method, while those associated with low-activity channels are simulated using an exact method. This path simulation technique is amenable for coupled path generation and a corresponding multilevel Monte Carlo algorithm. To estimate expected values of observables of the system at a prescribed final time...

Influence of fluid flow and heat transfer on crack propagation in SOFC multi-layered like material with anisotropic porous layers

An advanced numerical model based on the eXtended Finite Element Method (XFEM) is developed and used to investigate the effect of heat transfer and fluid flow on crack propagation in multi-layered SOFC with anisotropic porous layers. Effect of the anisotropy on the onset and the propagation path of cracks are investigated. Heat transport in the porous media is described by heat convection in the fluid phase, and heat conduction in both the fluid and solid phases. A time splitting technique is used to allow different numerical treatment for the heat transfer processes within the porous layer. A combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods are used to solve the advection–diffusion heat transfer in the flow channel and the fluid phase in the porous material. XFEM is used to solve the heat diffusion within the porous layers and the thermomechanical problem. The fluid flow, governed by Navier–Stokes and Darcy's law, is discretised with the nonconforming Crouzeix–Raviart (CR) finite element method. The multi-physics model is implemented and validated by comparing the computed Stress Intensity Factors (SIF) to ones from the literature. The methodology seems to be robust, accurate and the computational cost is reduced thanks to the XFEM.

An error evaluation on the vertical velocity algorithm in POM

A time splitting technique is common to many free surface ocean models. The different truncation errors in the equations of the internal and external modes require a numerical adjustment to make sure that algorithms correctly satisfy continuity equations and conserve tracers quantities. The princeton ocean model (POM) has applied a simple method of adjusting the vertical mean of internal velocities to external velocities at each internal time step. However, due to the Asselin time filter method adopted to prevent the numerical instability, the method of velocity adjustment used in POM can no longer guarantee the satisfaction of the continuity equation in the internal mode, though a special treatment is used to relate the surface elevation of the internal mode with that of the external mode. The error is proved to be a second-order term of the coefficient in the Asselin filter. One influence of this error in the numerical model is the failure of the kinetic boundary condition at the sea floor. By a regional experiment and a quasi-global experiment, the magnitudes of this error are evaluated, and several sensitivity tests of this error are performed. The characteristic of this error is analyzed and two alternative algorithms are suggested to reduce the error.

Influence of heat transfer and fluid flow on crack growth in multilayered porous/dense materials using XFEM: Application to Solid Oxide Fuel Cell like material design

An advanced numerical model is developed to investigate the influence of heat transfer and fluid flow on crack propagation in multi-layered porous materials. The fluid flow, governed by the Navier–Stokes and Darcy’s law, is discretized with the nonconforming Crouzeix–Raviart (CR) finite element method. A combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods is used to solve the advection–diffusion heat transfer equation in the flow channel and in the fluid phase within the porous material. The crack is assumed to affect only the heat diffusion within the porous layer, therefore a time splitting technique is used to solve the heat transfer in the fluid and the solid phases separately. Thus, within the porous material, the crack induces a discontinuity of the temperature at the crack surfaces and a singularity of the flux at the crack tip. Conduction in the solid phase is solved using the eXtended Finite Element Method (XFEM) to better handle the discontinuities and singularities caused by the cracks. The XFEM is also used to solve the thermo-mechanical problem and to track the crack propagation. The multi-physics model is implemented then validated for the transient regime, this necessitated a post processing treatment in which, the stress intensity factors (SIF) are computed for each time step. The SIFs are then used in the crack propagation criterion and the crack orientation angle. The methodology seems to be robust accurate and the computational cost is reduced thanks to the XFEM.

Error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems

We study space and time discretizations for mixed formulations of parabolic problems. The spatial approximation is based on the multipoint flux mixed finite element method, which reduces to an efficient cell-centered pressure system on general grids, including triangles, quadrilaterals, tetrahedra, and hexahedra. The time integration is performed by using a domain decomposition time-splitting technique combined with multiterm fractional step diagonally implicit Runge–Kutta methods. The resulting scheme is unconditionally stable and computationally efficient, as it reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel without the need for Schwarz-type iteration. Convergence analysis for both the semidiscrete and fully discrete schemes is presented.

Numerical Methods and Simulations for the Dynamics of One-Dimensional Zakharov–Rubenchik Equations

In this paper, we propose and study several accurate numerical methods for solving the one-dimensional Zakharov–Rubenchik equations (ZRE). We begin with a review on the important properties of the ZRE, including the solitary wave solutions and the various conservation laws. Then we propose a very efficient and accurate numerical method based on the time-splitting technique and the Fourier pseudo-spectral (TSFP) method. Next, we propose some conservative and non-conservative types of finite difference time domain methods, including a Crank–Nicolson finite difference method that conserves the mass and the energy of the system in the discrete level. Discrete conservation laws and numerical stability of all the proposed methods are analyzed. Comparisons between different methods in the efficiency, stability and accuracy are carried out, which identifies that the TSFP method is the most efficient and accurate numerical method among all the methods. Lastly, we apply the TSFP method to simulate and study the dynamics of the solitons in the ZRE numerically.

Kinetic description of ionospheric outflows based on the exact form of Fokker‐Planck collision operator: Electrons

We present the results of a finite difference implementation of the kinetic Fokker‐Planck model with an exact form of the nonlinear collisional operator. The model is time dependent and three‐dimensional; one spatial dimension and two in velocity space. The spatial dimension is aligned with the local magnetic field, and the velocity space is defined by the magnitude of the velocity and the cosine of pitch angle. An important new feature of model, the concept of integration along the particle trajectories, is discussed in detail. Integration along the trajectories combined with the operator time splitting technique results in a solution scheme which accurately accounts for both the fast convection of the particles along the magnetic field lines and relatively slow collisional process. We present several tests of the model's performance and also discuss simulation results of the evolution of the plasma distribution for realistic conditions in Earth's plasmasphere under different scenarios.

Performance analysis of parallel Schwarz preconditioners in the LES of turbulent channel flows

We present a comparative study of parallel Schwarz preconditioners in the solution of linear systems arising in a Large Eddy Simulation (LES) procedure for turbulent plane channel flows. This procedure applies a time-splitting technique to suitably filtered Navier–Stokes equations, in order to decouple the continuity and momentum equations, and uses a semi-implicit scheme for time integration and finite volumes for space discretisation. This approach requires the solution of four sparse linear systems at each time step, accounting for a large part of the overall simulation; hence the linear system solvers are a crucial component in the whole procedure. Several preconditioners are applied in the simulation of a reference test case for the LES community, using discretisation grids of different sizes, with the aim of analysing the effects of different algorithmic choices defining the preconditioners, and identifying the most effective ones for the selected problem. The preconditioners, coupled with the GMRES method, are run within SParC-LES, a recently developed LES code based on the PSBLAS and MLD2P4 libraries for parallel sparse matrix computations and preconditioning.

A time splitting algorithm for numerical solution of Richard’s equation

► Advection and diffusion forming the Richard’s equation are studied separately. ► A non-iterative numerical algorithm is proposed based on time splitting technique. ► A combined finite volume and characteristic methods are proposed for advection. ► To preserve the efficiency, an implicit procedure is proposed for characteristic part. ► A 4th order scheme is designed for diffusing without noticeable change in efficiency. The Richard’s equation mathematically expressing the infiltration problem is a transport type of equation describing two different phenomena namely advection and diffusion being hyperbolic and parabolic type of differential equation, respectively each having their own physical properties thus a single numerical scheme would not essentially be suitable for both. In this paper a numerical algorithm based on time splitting or fractional step method has been proposed and described. This has allowed designing and applying some different numerical schemes more compatible with the mathematical and physical properties of the corresponding phenomenon. In advection part an implicit characteristic based method has been employed to calculate the cell face fluxes and then the new unknowns have been obtained using finite volume method (FVM) explicitly. Two different speed type quantities namely wave celerity and mass velocity have been distinguished in the advection of infiltrated water to the soil, the magnitude of the first being several times larger than the other. The implicit characteristic based method has been designed specifically to cope with the mentioned high wave celerity. An implicit numerical scheme has been employed in which the space derivative term contributed in definition of cell face flux has been discretized in fourth order accurate manner. The extra cells getting involved due to the higher order discretization have been taken in account in a way not to change the tri-diagonal matrix coefficient shape preserving the efficiency of the algorithm. The performance, accuracy and efficiency of the out coming non-iterative numerical algorithm have been successfully examined by some test cases. The relative importance of the advection and diffusion terms in the concerned transport equation and the ration of their contribution in the overall infiltrated flux also have been discussed in detail.

Differential form and numerical implementation of Biot’s poroelasticity equations with squirt dissipation

The squirt-flow wave attenuation mechanism is implemented in Biot’s theory of poroelasticity in the form of differential equations. All the stiffnesses involved in the stress-strain relation become complex and frequency dependent, which can exactly be expressed in terms of kernels based on the Zener mechanical model. In the time domain, this approach implies time convolutions, which are circumvented by introducing memory variables. The differential equations are consistent with Gassmann’s and Mavko-Jizba equations at low and high frequencies, respectively. All the coefficients in the poro-viscoelastic differential equations have a clear physical meaning and can be obtained or estimated from independent measurements. The key additional parameters are the dry-rock bulk modulus at a confining pressure where all the compliant pores are closed, i.e., a hypothetical rock without the soft porosity, the grain-contact aspect ratio and the compliant porosity. We recasted the wave equation in the particle-velocity/stress formulation and solved it by using a time-splitting technique and the Fourier pseudospectral method to compute the spatial derivatives. The algorithm can be used to obtain synthetic wave fields in inhomogeneous media.

ADAPTIVE TIME SPLITTING METHOD FOR MULTI-SCALE EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS

This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady problems. The strategy considers a second order Strang method and another lower order embedded splitting scheme that takes into account potential loss of order due to the stiffness featured by time-space multi-scale phenomena. The scheme is then built upon a precise numerical analysis of the method and a complementary numerical procedure, conceived to overcome classical restrictions of adaptive time stepping schemes based on lower order embedded methods, whenever asymptotic estimates fail to predict the dynamics of the problem. The performance of the method in terms of control of integration errors is evaluated by numerical simulations of stiff propagating waves coming from nonlinear chemical dynamics models as well as highly multi-scale nanosecond repetitively pulsed gas discharges, which allow to illustrate the method capabilities to consistently describe a broad spectrum of time scales and different physical scenarios for consecutive discharge/post-discharge phases.