Strang Splitting Scheme Research Articles

A second-order Strang splitting scheme for the generalized Allen–Cahn type phase-field crystal model with FCC ordering structure

In this paper, we consider the generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure (PFC-FCC). Due to the combined complexity of the eighth-order spatial derivative and inherent nonlinearity, it poses a significant challenge to design a numerical scheme of high accuracy, stability, and efficiency to solve the PFC-FCC model. Endeavoring towards this objective, we adopt operator splitting method to address the PFC-FCC model. Based on Fourier spectral method for spatial discretization and SSP-RK method for temporal discretization, we propose an effective and easy-to-implement second-order scheme. We are also able to proffer an analytical proof for discrete mass conservation, and engage a discussion on an optimal error estimate for the second-order scheme. In addition, the energy stability of the numerical scheme is derived. Ultimately, a series of numerical experiments specifically designed to substantiate its accuracy, efficiency, and capability for phase transition are performed.

Comparison of two different integration methods for the (1+1)-dimensional Schrödinger-Poisson equation

We compare two different numerical methods to integrate in time spatially delocalized initial densities using the Schrödinger-Poisson equation system as the evolution law. The basic equation is a nonlinear Schrödinger equation with an auto-gravitating potential created by the wave function density itself. The latter is determined as a solution of Poisson's equation modelling, e.g., non-relativistic gravity. For reasons of complexity, we treat a one-dimensional version of the problem whose numerical integration is still challenging because of the extreme long-range forces (being constant in the asymptotic limit). Both of our methods, a Strang splitting scheme and a basis function approach using B-splines, are compared in numerical convergence and effectivity. Overall, our Strang-splitting evolution compares favourably with the B-spline method. In particular, by using an adaptive time-stepper rather large one-dimensional boxes can be treated. These results give hope for extensions to two spatial dimensions for not too small boxes and large evolution times necessary for describing, for instance, dark matter formation over cosmologically relevant scales.

A second-order Strang splitting scheme with exponential integrating factor for the Allen–Cahn equation with logarithmic Flory–Huggins potential

In this paper, we mainly consider the numerical approximation for the Allen-Cahn(AC) equation with logarithmic Flory–Huggins potential. It is well-known that the logarithmic Flory–Huggins potential is more physically significant than the generally employed fourth-order polynomial potential in various phase-field models. Nevertheless, owing to the singularity of the logarithmic term and the strong nonlinearity of the energy potential function, it is extremely challenging to develop an effective and easy-to-implement numerical scheme that preserves the maximum principle and energy dissipation law. To overcome these difficulties, we construct a second-order Strang splitting scheme with exponential integrating factor. Firstly, base on the idea of dimensional splitting, a new time dependent function, which is called exponential integrating factor is introduced. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the logarithmic term is treated by explicit 2-stage strong stability preserving Runge–Kutta(SSP-RK2) method. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence in details for the proposed scheme. A variety of numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method.

Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

A Robust Reacting Flow Solver with Computational Diagnostics Based on OpenFOAM and Cantera

In this study, we developed a new reacting flow solver based on OpenFOAM (OF) and Cantera, with the capabilities of (i) dealing with detailed species transport and chemistry, (ii) integration using a well-balanced splitting scheme, and (iii) two advanced computational diagnostic methods. First of all, a flaw of the original OF chemistry model to deal with pressure-dependent reactions is fixed. This solver then couples Cantera with OF so that the robust chemistry reader, chemical reaction rate calculations, ordinary differential equations (ODEs) solver, and species transport properties handled by Cantera can be accessed by OF. In this way, two transport models (mixture-averaged and constant Lewis number models) are implemented in the coupled solver. Finally, both the Strang splitting scheme and a well-balanced splitting scheme are implemented in this solver. The newly added features are then assessed and validated via a series of auto-ignition tests, a perfectly stirred reactor, a 1D unstretched laminar premixed flame, a 2D counter-flow laminar diffusion flame, and a 3D turbulent partially premixed flame (Sandia Flame D). It is shown that the well-balanced property is crucial for splitting schemes to accurately capture the ignition and extinction events. To facilitate the understanding on combustion modes and complex chemistry in large scale simulations, two computational diagnostic methods (conservative chemical explosive mode analysis, CCEMA, and global pathway analysis, GPA) are subsequently implemented in the current framework and used to study Sandia Flame D for the first time. It is shown that these two diagnostic methods can extract the flame structure, combustion modes, and controlling global reaction pathways from the simulation data.

Simulation of three-dimensional forced compressible isotropic turbulence by a redesigned discrete unified gas kinetic scheme

In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions

The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov–Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang splitting scheme is introduced that retains second-order accuracy. For this numerical scheme a stability analysis is conducted. In addition, numerical results are shown to support the theoretical derivations.

Optimal convergence of a second-order low-regularity integrator for the KdV equation

AbstractIn this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An exponential-type integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides second-order accuracy in $H^\gamma $ for initial data in $H^{\gamma +4}$ for any $\gamma \geq 0$, where the regularity requirement is lower than for classical methods. The result is confirmed by numerical experiments, and comparisons are made with the Strang splitting scheme.

Discrete unified gas kinetic scheme simulation of conjugate heat transfer problems in complex geometries by a ghost-cell interface method

In this paper, we extend the discrete unified gas kinetic scheme (DUGKS) to simulate conjugate heat transfer flows. The conjugate interface that separates two media with different thermophysical properties is handled by a ghost-cell (GC) immersed boundary method. Specifically, fictitious cells are set in both domains to implement the continuity conditions of both temperature and heat flux at the conjugate interface. Information at the ghost cells is unavailable and should be reconstructed from the interface and neighboring field. With interface constraints involved in the reconstruction, the conjugate condition can be incorporated smoothly into the process of solving the full flow field. Those features make the present GC-DUGKS capture the conjugate interface, without introducing an additional term or modifying the equilibrium distribution function as in previous studies. Furthermore, the Strang-splitting scheme is applied for convenient handling of the thermal force term in the governing equation. Simulations of several well-established natural convection flows containing complex conjugate interfaces are performed to validate the GC-DUGKS. The results demonstrate the accuracy and feasibility of the present method for conjugate heat transfer problems.

A second-order dynamic adaptive hybrid scheme for time-integration of stiff chemistry

A dynamic adaptive hybrid integration (AHI) scheme of second-order accuracy (AHI2) is proposed for time-integration of chemically reacting flows involving stiff chemistry. AHI2 is extended from a first-order AHI method (AHI1) developed in a previous study, which showed that when significant radical sources are present in the non-chemical source terms, splitting the chemical and the transport sub-systems may incur O(1) errors unless the splitting time steps are comparable to or smaller than that required for explicit integration. As such, the transport term needs to be carried during the integration of stiff chemistry to avoid the large splitting errors. In AHI, fast species and reactions that may induce stiffness are treated implicitly, while the non-stiff variables and source terms, including slow reactions and the mixing term, are treated explicitly. The separation of fast-slow chemistry is performed on-the-fly based on analytically evaluated timescales for species and reactions, such that the complexity of the implicit core in the governing equations is minimized at each time step and the time-integration can be performed with high efficiency. The newly developed AHI2 scheme combines the midpoint scheme and the trapezoidal rule to achieve second-order accuracy. The second-order scheme is tested with a toy problem, as well as auto-ignition and unsteady perfectly stirred reactors (PSR) with detailed chemistry. Results show that AHI2 can significantly improve accuracy compared with AHI1. It was further found that AHI2 can accurately predict extinction of unsteady PSRs while the Strang splitting scheme fails to control the error, showing the necessity not to split the chemistry and transport source terms for prediction of extinction or forced-ignition problems involving significant radical sources. Further analysis of numerical efficiency shows that for auto-ignition AHI2 reduces computational cost primarily through the reduction in the number of variables to be solved implicitly, and the time-saving increases with the mechanism size, reaching approximately 70% for the 111-species USC-Mech II compared with a fully implicit scheme. For unsteady PSR involving homogeneous mixing, AHI2 achieved speedup factors of 20 to 30 compared with the Strang splitting scheme. Furthermore, sparse matrix techniques are integrated into AHI2 (AHI2-S) to achieve high computational efficiency. It is shown that the computational cost of AHI2-S is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. It is further shown that AHI2-S achieves a speed-up factor of around two compared with the efficient fully implicit sparse solver LSODES with analytic Jacobian.

Computing the finite time Lyapunov exponent for flows with uncertainties

We propose an Eulerian approach to compute the expected finite time Lyapunov exponent (FTLE) of uncertain flow fields. The definition extends the usual FTLE for deterministic dynamical systems. Instead of performing Monte Carlo simulations as in typical Lagrangian computations, our approach associates each initial flow particle with a probability density function (PDF) which satisfies an advection-diffusion equation known as the Fokker-Planck (FP) equation. Numerically, we incorporate Strang's splitting scheme so that we can obtain a second-order accurate solution to the equation. To further improve the computational efficiency, we develop an adaptive approach to concentrate the computation of the FTLE near the ridge, where the so-called Lagrangian coherent structure (LCS) might exist. We will apply our proposed algorithm to several test examples including a real-life dataset to demonstrate the performance of the method.

A ghost-cell discrete unified gas kinetic scheme for thermal flows with heat flux at curved interface

The discrete unified gas kinetic scheme (DUGKS) is advanced for simulating thermal convections with curved heat flux condition using the ghost-cell (GC) approach. The fluid flow and temperature field combined under the Boussinesq approximation are solved by the double-population model. The ghost-cell immersed boundary method is applied to the curved heat-flux interface, where fictitious cells are set in the solid domain. The information at the centers of those cells is extrapolated from the fluid-solid interface and the neighboring flow. The heat-flux boundary condition at the interface can then be incorporated into the solution of the entire thermal flow. Note that the extrapolation/interpolation involved in the GC-DUGKS is accomplished through a sharp interface manner. Therefore, the difficulty in handling the heat-flux boundary condition for the diffuse-type immersed boundary methods due to the cross contamination is removed. Furthermore, the thermal buoyancy force is conveniently combined with the governing equation by the Strang-splitting scheme. Simulations of several well-established convection-diffusion flow problems with heat flux at the curved interface are performed to validate the present GC-DUGKS. The results demonstrate the accuracy and feasibility of the method.

On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

Efficient exponential splitting spectral methods for linear Schrödinger equation in the semiclassical regime

The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrödinger equation in the semiclassical regime, where the Planck constant ε is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrödinger equation with a linear potential. The local discretization error of the two time-splitting methods constructed here is O(max⁡{Δt3,Δt5/ε}), while the well-known Lie-Trotter splitting scheme and the Strang splitting scheme are O(Δt2/ε) and O(Δt3/ε), respectively, where Δt is the time step-size. The global error estimates of new exponential splitting schemes with spectral discretization suggests that larger time step-size is admissible for obtaining high accuracy approximation of the solutions. Numerical studies verify our theoretical results and reveal that the new methods are especially efficient for linear semiclassical Schrödinger equation with a quadratic potential.

Splitting methods for nonlinear Dirac equations with Thirring type interaction in the nonrelativistic limit regime

Nonlinear Dirac equations describe the motion of relativistic spin-12 particles in presence of external electromagnetic fields, modelled by an electric and magnetic potential, and taking into account a nonlinear particle self-interaction. In recent years, the construction of numerical splitting schemes for the solution of these systems in the nonrelativistic limit regime, i.e., the speed of light c formally tending to infinity, has gained a lot of attention. In this paper, we consider a nonlinear Dirac equation with Thirring type interaction, where in contrast to the case of the Soler type nonlinearity a classical two-term splitting scheme cannot be straightforwardly applied. Thus, we propose and analyse a three-term Strang splitting scheme which relies on splitting the full problem into the free Dirac subproblem, a potential subproblem, and a nonlinear subproblem, where each subproblem can be solved exactly in time. Moreover, our analysis shows that the error of our scheme improves from Oτ2c4 to Oτ2c3 if the magnetic potential in the system vanishes. Furthermore, we propose an efficient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime c≫1 which allows errors of order Oc−1 without any c-dependent time step restriction.

Operator splitting method for numerical solution of modified equal width equation

In this manuscript, numerical solutions of the equations in the form of $u_{t}=Au+B(u)$ have been sought for, where $A$ and $B$ are linear and nonlinear operators, respectively. The modified equal width (MEW) equation has been converted into two sub problems. Then, the sub problems were solved according to the Strang splitting scheme by applying the cubic B-spline collocation finite element method. Thus, more accurate results of the equation MEW have been obtained than those non-splitting users. In order to test the accuracy and efficiency of the present method; single soliton, interaction of two solitons and Maxwellian initial condition pulse problems have been considered. Moreover, the stability analysis of each sub problem has been investigated by von-Neumann analysis method.

A non-iterative direct-forcing immersed boundary method for thermal discrete unified gas kinetic scheme with Dirichlet boundary conditions

A non-iterative immersed boundary-thermal discrete unified gas kinetic scheme (IB-TDUGKS) is developed for the simulation of thermal flows. Two sets of distribution functions are applied for the velocity and temperature fields respectively, where the influence of the heat on the fluid is considered by the Boussinesq approximation. The immersed boundary method with the direct-forcing model is used to handle the complex solid boundary, in which it is replaced by the generator of local body force and heat source/sink. However, the explicit force and heat source/sink in the conventional IB methods usually result in the unphysical fluid and heat penetrations. The iterative forcing approach can remove such deficiency, but it increases both the complexity of solutions and the computational load. Therefore, a non-iterative IB method is presented in this study. With the introduction of an adjustment parameter, the present approach evaluates the force and heat source/sink explicitly, and at the same time enforces the boundary conditions for both the velocity and temperature at the fluid-solid interface. Furthermore, the force and heat source/sink are conveniently incorporated into the DUGKS with the Strang-splitting scheme. The accuracy and feasibility of the present IB-TDUGKS is verified in several well-established thermal flow problems. The results agree well the data available in the literature.

A spectral radius scaling semi-implicit iterative time stepping method for reactive flow simulations with detailed chemistry

A spectral radius scaling semi-implicit time stepping scheme has been developed for simulating unsteady compressible reactive flows with detailed chemistry, in which the spectral radius in the LUSGS scheme has been augmented to account for viscous/diffusive and reactive terms and a scalar matrix is proposed to approximate the chemical Jacobian using the minimum species destruction timescale. The performance of the semi-implicit scheme, together with a third-order explicit Runge–Kutta scheme and a Strang splitting scheme, have been investigated in auto-ignition and laminar premixed and nonpremixed flames of three representative fuels, e.g., hydrogen, methane, and n-heptane. Results show that the minimum species destruction time scale can well represent the smallest chemical time scale in reactive flows and the proposed scheme can significantly increase the allowable time steps in simulations. The scheme is stable when the time step is as large as 10 μs, which is about three to five orders of magnitude larger than the smallest time scales in various tests considered. For the test flames considered, the semi-implicit scheme achieves second order of accuracy in time. Moreover, the errors in quantities of interest are smaller than those from the Strang splitting scheme indicating the accuracy gain when the reaction and transport terms are solved coupled. Results also show that the relative efficiency of different schemes depends on fuel mechanisms and test flames. When the minimum time scale in reactive flows is governed by transport processes instead of chemical reactions, the proposed semi-implicit scheme is more efficient than the splitting scheme. Otherwise, the relative efficiency depends on the cost in sub-iterations for convergence within each time step and in the integration for chemistry substep. Then, the capability of the compressible reacting flow solver and the proposed semi-implicit scheme is demonstrated for capturing the hydrogen detonation waves. Finally, the performance of the proposed method is demonstrated in a two-dimensional hydrogen/air diffusion flame.

Semi-implicit iterative methods for low Mach number turbulent reacting flows: Operator splitting versus approximate factorization

Two formally second-order accurate, semi-implicit, iterative methods for the solution of scalar transport–reaction equations are developed for Direct Numerical Simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an approximately factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal approximation to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the approximately factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame.

A sparse stiff chemistry solver based on dynamic adaptive integration for efficient combustion simulations

A sparse stiff chemistry solver based on dynamic adaptive hybrid integration (AHI-S) is developed and demonstrated for efficient combustion simulations. In a previous study, a dynamic adaptive method for hybrid integration (AHI) was developed to speed up the time integration of chemically reacting flows with detailed chemistry. The AHI method solves the fast subcomponent of chemistry implicitly and the slow subcomponent of chemistry and transport explicitly, and it was shown that AHI is more accurate and efficient than the operator-splitting schemes when there are significant radical sources from the transport term. In the present study, the AHI method is first improved to minimize the number of nontrivial entries in the Jacobian. Sparse matrix techniques are further integrated into AHI to achieve high computational efficiency. The performance of the new AHI-S solver is investigated in constant-pressure auto-ignition systems using different mechanisms that consist of 9–2878 species. It is shown that the computational cost of the AHI-S solver is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. The AHI-S solver achieves speed-up factors ranging from approximately 10, for the 9-species hydrogen mechanism, to approximately 3000, for the 2878-species biodiesel mechanism, compared with the fully implicit VODE solver with Jacobian evaluated through numerical perturbations and factorized with dense matrix operations. It is further found that for mechanisms with less than approximately 100 species, the time saving of AHI-S is primarily attributed to the reduced size of the implicit core of the governing equations, while for mechanisms with more than 100 species, the computational cost of VODE is dominated by the dense LU factorization, such that the time saving of AHI-S is mostly attributed to the sparse LU factorization. The AHI-S solver is then applied to unsteady perfectly stirred reactors involving extinction and re-ignition. Speed-up factors from 50 to 30,000 are achieved compared with the Strang splitting scheme with the chemistry substeps implicitly integrated with VODE, while speed-up factors of 10–100 are achieved compared with the Strang splitting scheme implemented with the sparse stiff LSODES solver. In the end, the performance of AHI-S is investigated in one-dimensional (1-D) unsteady freely propagating laminar premixed flames for a methane/air mixture, for which the time step size in AHI-S is limited by the fastest transport process. A speed-up factor of approximately 200 is achieved compared with the Strang splitting scheme for fixed time step sizes between 10−8s and 10−6s.