Time Step Size Research Articles

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

Large-scale surface hopping simulation of charge transport in hexagonal molecular crystals: role of electronic coupling signs

In large-scale surface hopping simulations with a huge number of electronic states, trivial crossings could easily lead to incorrect long-range charge transfer and induce large numerical errors. We here study the charge transport in two-dimensional hexagonal molecular crystals with a parameter-free full crossing corrected global flux surface hopping method. Fast time-step size convergence and system size independence have been realized in large systems containing thousands of molecular sites. In hexagonal systems, each molecular site has six nearest neighbours. We find that the signs of their electronic couplings have a strong impact on the charge mobility and delocalization strength. In particular, changing the signs of electronic couplings can even lead to a transition from hopping to band-like transport. In comparison, such phenomena cannot be observed in extensively studied two-dimensional square systems. This is attributed to symmetry of the electronic Hamiltonian and distribution of the energy levels. Due to its high performance, the proposed approach is promising to be applied to more realistic and complex systems for molecular design.

Explicit asynchronous time scheme with local push-forward stepping for discontinuous elastic wave propagation: One-dimensional heterogeneous cases and Hopkinson bar experiment

This is a presentation of robust and accurate explicit time-stepping strategy for finite element modeling of elastic discontinuous wave propagation in strongly heterogeneous, multi-material and graded one-dimensional media. One of the major issues in FEM modeling is the existence of spurious numerical stress oscillations close to theoretical wave fronts due to temporal-spatial dispersion behavior of FE discretization. The numerical strategy presented for modeling of 1D discontinuous elastic waves is based on (a) pushforward-pullback local stepping — ensuring the elimination of dispersion due to different critical time step sizes of finite elements, (b) domain decomposition via localized Lagrange multipliers — to satisfy coupling kinematics and dynamic equations , (c) asynchronous time scheme — ensuring the correct information transfer of quantities for the case of integer ratios of time step size for all domain pairs. Dispersion behaviors, existence of spurious stress oscillations, and sensitivity of the dispersion to time step size are then suppressed. The proposed method is numerically tested with regard to the rectangular step pulse elastic propagation problem considering in-space varying Young’s modulus. To prove robustness and accuracy, a comparison with results from commercial software, an analytical solution, and experimental data from partial assembly of a split Hopkinson pressure bar (SHPB) setup is provided.

An unconditionally energy-stable scheme for the convective heat transfer equation

PurposeThis paper aims to present an unconditionally energy-stable scheme for approximating the convective heat transfer equation.Design/methodology/approachThe scheme stems from the generalized positive auxiliary variable (gPAV) idea and exploits a special treatment for the convection term. The original convection term is replaced by its linear approximation plus a correction term, which is under the control of an auxiliary variable. The scheme entails the computation of two temperature fields within each time step, and the linear algebraic system resulting from the discretization involves a coefficient matrix that is updated periodically. This auxiliary variable is given by a well-defined explicit formula that guarantees the positivity of its computed value.FindingsCompared with the semi-implicit scheme and the gPAV-based scheme without the treatment on the convection term, the current scheme can provide an expanded accuracy range and achieve more accurate simulations at large (or fairly large) time step sizes. Extensive numerical experiments have been presented to demonstrate the accuracy and stability performance of the scheme developed herein.Originality/valueThis study shows the unconditional discrete energy stability property of the current scheme, irrespective of the time step sizes.

High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Padé expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius ρ∞ in the high frequency limit. Using this user-specified parameter as a weight factor, a Padé expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient time-stepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.

An energy stable incompressible SPH method with consistent solid boundary treatment

We propose two novel solid wall boundary treatments on the energy-stable Smoothed Particle Hydrodynamics (SPH) method for the Navier–Stokes system which models single-phase incompressible flows. In this work, the solid wall is discretized with dummy particles, and an external force is introduced to restrict the motion of solid particles, whose physical significance and numerical necessity are analyzed. We construct consistent fluid–solid coupling schemes based on the approaches so that the boundary condition is involved in the same equations system with particular constraints. Furthermore, the schemes are unconditionally energy-stable, which allows large time step sizes and leads to efficient computation. A variety of examples are demonstrated to verify the corresponding numerical analysis result, which shows that the approaches can easily handle no-slip and moving boundaries. Finally, we highlight that the energy-stable ISPH with consistent boundary treatments greatly benefits solving long-term problems with steady states.

A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions

AbstractThis article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of with time step and mesh size in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete norm or norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.

An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models

In this paper, we propose and analyze a variant of Lagrange multiplier approach with a penalty term for phase-field models. More precisely, we develop two types of linear second-order decoupled and unconditionally energy stable time-stepping schemes for phase-field models based on the Crank-Nicolson (CN) and the second order backward differentiation formulations (BDF2), where the scalar multiplier variable is evaluated by solving a nonlinear algebraic equation. Comparing with the Lagrange multiplier method (Cheng et al. (2020) [6]) for phase-field models, a scalar penalty term is first introduced to improve the time-step size constraint for the convergence of Newton's iterative algorithm for the nonlinear algebraic equation, which allows us to obtain robust and efficient numerical schemes for phase-field models. Moreover, we only use first order time-stepping scheme to approximate the scalar multiplier variable, which will not affect the second order accuracy of the unknown phase function with some special Lagrange multiplier functions. Meanwhile, it plays an essential role in the derivation of energy stability of the proposed BDF2-based scheme on the nonuniform temporal mesh. In addition, the energy stability of the CN scheme is rigorously established. Furthermore, we deduce that the proposed BDF2 scheme is energy stable with the adjacent time-step ratios no more than 4.8645. Finally, a series of numerical tests are presented to numerically validate the efficiency and stability of our proposed Lagrange multiplier approach.

High fidelity transient solver in STREAM based on multigroup coarse-mesh finite difference method

This study incorporates a high-fidelity transient analysis solver based on multigroup CMFD in the MOC code STREAM. Transport modeling with heterogeneous geometries of the reactor core increases computational cost in terms of memory and time, whereas the multigroup CMFD reduces the computational cost. The reactor condition does not change at every time step, which is a vital point for the utilization of CMFD. CMFD correction factors are updated from the transport solution whenever the reactor core condition changes, and the simulation continues until the end. The transport solution is adjusted once CMFD achieves the solution. The flux-weighted method is used for rod decusping to update the partially inserted control rod cell material, which maintains the solution's stability. A smaller time-step size is needed to obtain an accurate solution, which increases the computational cost. The adaptive step-size control algorithm is robust for controlling the time step size. This algorithm is based on local errors and has the potential capability to accept or reject the solution. Several numerical problems are selected to analyze the performance and numerical accuracy of parallel computing, rod decusping, and adaptive time step control. Lastly, a typical pressurized LWR was chosen to study the rod-ejection accident.

Ergodic estimators of double exponential Ornstein–Uhlenbeck processes

The goal of this paper is to construct ergodic estimators for double exponential Ornstein–Uhlenbeck process, where the process is observed at discrete time instants with time step size h. We show the existence and uniqueness of the function equations to determine the estimators for fixed time step size h. Also, we show the strong consistency and the asymptotic normality of the estimators. Furthermore, we propose a simulation method of the double exponential Ornstein–Uhlenbeck process and perform some numerical simulations to demonstrate the effectiveness of the proposed estimators.

A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete H 1 H^{1} error estimate and energy stability for the classic constant mobility case and the L ∞ L^{\infty } error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.

Lack of robustness and accuracy of many numerical schemes for phase-field simulations

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size [Formula: see text], there exists an initial condition [Formula: see text], with [Formula: see text], such that the numerical solution converges to the wrong steady state solution. (iv) For [Formula: see text], all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.

Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche’s method. The conventional stability requirement for Nitsche’s method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff–Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

A novel approach of unconditional optimal error estimate of linearized and conservative Galerkin FEM for Klein–Gordon–Schrödinger equations

This paper is devoted to the unconditional optimal error analysis of a linearized, decoupled and conservative Galerkin finite element method (FEM) for the Klein–Gordon–Schödinger equations. The present work has two main contributions: One is that the mass and energy conservation properties of the scheme are proved, and a priori boundedness of the numerical solutions is derived. The other one is that a novel approach is developed and analyzed, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size required in the previous works, and may lead to the unconditional optimal error estimates for the related variables. Some numerical results are also provided to confirm the theoretical analysis and show the efficiency of the proposed scheme.

Efficient inequality-preserving integrators for differential equations satisfying forward Euler conditions

Developing explicit, high-order accurate, and stable algorithms for nonlinear differential equations remains an exceedingly difficult task. In this work, a systematic approach is proposed to develop high-order, large time-stepping schemes that can preserve inequality structures shared by a class of differential equations satisfying forward Euler conditions. Strong-stability-preserving (SSP) methods are popular and effective for solving equations of this type. However, few methods can deal with the situation when the time-step size is larger than that allowed by SSP methods. By adopting time-step-dependent stabilization and taking advantage of integrating factor methods in the Shu–Osher form, we propose enforcing the inequality structure preservation by approximating the exponential function using a novel recurrent approximation without harming the convergence. We define sufficient conditions for the obtained parametric Runge–Kutta (pRK) schemes to preserve inequality structures for any time-step size, namely, the underlying Shu–Osher coefficients are non-negative. To remove the requirement of a large stabilization term caused by stiff linear operators, we further develop inequality-preserving parametric integrating factor Runge–Kutta (pIFRK) schemes by incorporating the pRK with an integrating factor related to the stiff term, and enforcing the non-decreasing of abscissas. The only free parameter can be determined a priori based on the SSP coefficient, the time-step size, and the forward Euler condition. We demonstrate that the parametric methods developed here offer an effective and unified approach to study problems that satisfy forward Euler conditions, and cover a wide range of well-known models. Finally, numerical experiments reflect the high-order accuracy, efficiency, and inequality-preserving properties of the proposed schemes.

DoD Stabilization of linear hyperbolic PDEs on general cut‐cell meshes

AbstractStandard numerical methods for hyperbolic PDEs require for stability a CFL‐condition which implies that the time step size depends on the size of the elements of the mesh. On cut‐cell meshes, elements can become arbitrarily small and thus the time step size cannot take the size of small cut‐cells into account but has to be chosen based on the background mesh elements.A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical and numerical properties have been shown in one and two space dimensions [4, 9]. Up to now the method is restricted to stabilization of cut‐cells with exactly one inflow and one outflow face, i.e. triangular cut‐cells with a no‐flow face (see [4]).We extend the DoD stabilization to cut‐cells with multiple in‐ and outflow faces by properly considering the flow distribution inside the cut‐cell. We further prove L2‐stability for the semi‐discrete formulation in space and present numerical results to validate the proposed extension.

Finite-Element A-V Formulation for EMQS Problems via Two-Domain Continuity Gauging

A finite-element A-V formulation is developed to solve EMQS (electromagnetic quasistatic), aka Darwin, problems in the time and frequency domains by making use of a two-domain continuity gauging strategy. The formulation utilizes the magnetic vector and the electric scalar potentials, and an additional scalar in the conducting regions to make the discretized Galerkin system symmetric. The resulting FE (finite element) matrix is LF (low-frequency) stable in the frequency domain. As a result, the time-step size is not limited by the characteristics of the system matrix in the backward Euler time-stepping scheme. The system matrixes are also regular, which allows choosing a direct solver to speed up the temporal solution significantly. The formulation is verified in both the frequency and the time domains by solving benchmark problems, and the results are compared to full-wave and eddy-current solutions.

A Novel Energy and Momentum Consistent Mixed Formulation for Coupled Nonlinear Electro‐Thermo‐Elastodynamics

AbstractIn the present contribution, a novel mixed formulation aimed at the energy‐momentum consistent simulation of coupled nonlinear electro‐thermo‐elastodynamic systems, in particular dielectric elastomer actuators, is proposed. It is essentially based on a mixed framework for elastodynamics in the case of polyconvex stored energy functions. In accordance with this framework, the properties of the rediscovered tensor cross product are exploited in a first step to derive a mixed formulation via a Hu‐Washizu type extension of the stored energy function. Afterwards, the corresponding strong form is derived and supplemented with the (initial) boundary value problems of thermodynamics and electrostatics. By additionally choosing an appropriate polyconvexity‐inspired energy density function, this procedure leads to a fully coupled electro‐thermo‐elastodynamic formulation that benefits from the properties of the underlying mixed framework.Furthermore, an energy‐momentum consistent time integration scheme is proposed for the novel framework, where discrete derivatives in the sense of Gonzalez are employed. The formulation is second‐order accurate and stable even for large time step sizes. Eventually, the performance of the novel formulation is illustrated in a numerical example.