Allowable Time Step Research Articles

Surprisingly tight Courant–Friedrichs–Lewy condition in explicit high-order Arakawa schemes

The Arakawa scheme, which exactly conserves energy and enstrophy in two-dimensional hydrodynamics, severely reduces the maximum allowable stable time step compared to the Courant–Friedrichs–Lewy condition when combined with a spectral or spectral-like high-order discretization. We calculate the time step restriction by finding the responsible high frequency eigenfunctions and suggest a remedy. The phenomenon has relevance for analogous methods conserving multiple quadratic invariants in gyrokinetic or magnetohydrodynamic simulations.

CFL Optimized Forward–Backward Runge–Kutta Schemes for the Shallow-Water Equations

Abstract We present the formulation and optimization of a Runge–Kutta-type time-stepping scheme for solving the shallow-water equations, aimed at substantially increasing the effective allowable time step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second-order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality. Significance Statement The purpose of this work is to develop and optimize time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically, for the shallow-water equations we optimize for schemes that can take time steps as large as possible while retaining solution quality. We find that our optimized schemes can take time steps between 1.6 and 2.2 times larger than schemes that cost the same number of floating point operations, translating directly to a corresponding speedup. Our ultimate goal is to use these schemes in climate-scale simulations.

Multiblock parallel high-order implicit residual smoothing time scheme for compressible Navier–Stokes equations

In direct and large eddy simulations, very small space steps are used close to the solid walls in order to resolve the boundary-layer structures. Due to the restrictive CFL stability criteria of explicit time-stepping schemes, the maximum allowable time step is also very small, leading to high computational costs, notably for converging flow statistics. The use of an implicit integration scheme may overcome this limitation at the price of an increased computational cost per step. Moreover, the most commonly used fully implicit schemes induce higher errors due to the necessary approximations and poor dispersion and dissipation properties. As a compromise, a fourth-order implicit residual smoothing scheme (IRS4), successfully validated for a finite volume solver in Cinnella nad Content (2016); Hoarau and Cinnella (2020), has been introduced in a multiblock high-order finite-difference solver. Several improvements are proposed to ensure better dissipation properties, a more efficient treatment of physical boundaries and an accurate and stable parallel multiblock implementation. For moderate CFL numbers, a similar accuracy as the explicit method is obtained with substantial savings in terms of computational time.

On the Stability of IMEX Upwind gSBP Schemes for 1D Linear Advection-Diffusion Equations

AbstractA fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts (upwind gSBP) schemes in space and implicit-explicit Runge-Kutta (IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly, while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first- and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin (DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay (BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space, since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.

Speeding up mesoscale thermal simulations of powder bed additive manufacturing thanks to the forward Euler time-integration scheme: A critical assessment

This work addresses the mesoscale conduction-based modelling of Powder Bed Fusion for metallic materials. The forward Euler (explicit) time integration strategy is thoroughly benchmarked for the first time in a Powder Bed Fusion simulation. We also compare the computational times of explicit and implicit schemes when applied to an experimentally validated Powder Bed Fusion simulation. We demonstrate that the use of the former scheme results in considerable speedups (20 to 100) without noticeable loss of accuracy. We furthermore propose a Powder Bed Fusion-dedicated procedure to estimate the maximum allowable time step size of the explicit scheme.

Analysis and design of a local time stepping scheme for LES acceleration in reactive and non-reactive flow simulations

Explicit time integration based CFD solvers suffer from restriction on the maximum allowable time step computed from the well known Courant-Friedrichs-Lewy (CFL) stability criterion. This restriction poses severe challenge in carrying out large eddy simulation (LES) of reactive and non-reactive flows, where the grid resolution is fine. The challenge of restricted time step is further augmented when dealing with large computational domains that pose a wide disparity in the system time scales. In this study, a numerical methodology is presented based on local time stepping in an overset grid framework. The attainable speedup is found to be a function of the ratio of time steps used in the sub-domains and the ratio of the number of computational degrees of freedom. The method is analyzed using global spectral analysis (GSA) and shows excellent agreement in solution accuracy with the conventional explicit time integration based solver. The impact of local time stepping on the order of accuracy and global conservation properties are also presented. This method is then applied to simulate three flow test cases to demonstrate the ability of the method to reproduce the first and second-order turbulent statistics at reduced computational time.

Precise-Integration Time-Domain Formulation for Optical Periodic Media.

A numerical formulation based on the precise-integration time-domain (PITD) method for simulating periodic media is extended for overcoming the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the periodic boundary conditions are implemented, permitting the simulation of a wide range of periodic optical media, i.e., gratings, or thin-film filters. Furthermore, the complete tensorial derivation for the permittivity also allows simulating anisotropic periodic media. Numerical results demonstrate that PITD is reliable and even considering anisotropic media can be competitive compared to traditional FDTD solutions. Furthermore, the maximum allowable time-step size has been demonstrated to be much larger than that of the CFL limit of the FDTD method, being a valuable tool in cases in which the steady-state requires a large number of time-steps.

A prominent unified crosstalk model for linear and sub-threshold regions in mixed CNT bundle interconnects

With the feasibility to scale the devices and interconnects in highly sophisticated VLSI technology, the demand for high-speed and low-power e-applications have also subsequently increased stupendously. Based on the applications and requirements, a VLSI system is operated in different modes as linear or sub-threshold. A unified analytical model for describing both these linear and sub-threshold modes is highly desirable. This has been innovatively presented in the current paper. Moreover, the futuristic and emerging mixed carbon nanotube bundle (MCB) as interconnects for both linear and sub-threshold region of operations has been considered. The essential signal integrity analysis comprising of several crosstalk effects such as associated transient response, delay and power have been analyzed. To evaluate this, analytical model is formulated and proposed using accurate unconditionally-stable finite-difference time-domain (USFDTD) technique. Utilizing the proposed model, it is investigated that linear mode of operation is good for realizing high-speed systems while sub-threshold is a preferable operation for applications targeted for low-power. Comprehensively, it is envisaged that the average power-delay-product in MCB interconnects operating in sub-threshold region is low and reduced by 74% compared to corresponding linear region of operation. Also, it is demonstrated that the proposed unified USFDTD based model for MCB interconnects operating in different modes of operation is stable and not constricted by the Courant condition. At maximum allowable time step, the proposed model is nearly 10 and 25 times faster than the conventional FDTD analytical and HSPICE simulation models, respectively. The results reveal that USFDTD technique provides better accuracy than the FDTD technique. The different performance analyses are performed at 22 nm technology node.

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

An arbitrary high-order discontinuous Galerkin method with local time-stepping for linear acoustic wave propagation.

This paper presents a numerical scheme of arbitrary order of accuracy in both space and time, based on the arbitrary high-order derivatives methodology, for transient acoustic simulations. The scheme combines the nodal discontinuous Galerkin method for the spatial discretization and the Taylor series integrator (TSI) for the time integration. The main idea of the TSI is a temporal Taylor series expansion of all unknown acoustic variables in which the time derivatives are replaced by spatial derivatives via the Cauchy-Kovalewski procedure. The computational cost for the time integration is linearly proportional to the order of accuracy. To increase the computational efficiency for simulations involving strongly varying mesh sizes or material properties, a local time-stepping (LTS) algorithm accompanying the arbitrary high-order derivatives discontinuous Galerkin (ADER-DG) scheme, which ensures correct communications between domains with different time step sizes, is proposed. A numerical stability analysis in terms of the maximum allowable time step sizes is performed. Based on numerical convergence analysis, we demonstrate that for nonuniform meshes, a consistent high-order accuracy in space and time is achieved using ADER-DG with LTS. An application to the sound propagation across a transmissive noise barrier validates the potential of the proposed method for practical problems demanding high accuracy.

Accelerated implicit-explicit Runge-Kutta schemes for locally stiff systems

In this paper we introduce a family of accelerated implicit-explicit (AIMEX) schemes for the solution of stiff systems of equations. Similar to conventional IMEX schemes, AIMEX schemes allow a problem to be split into its stiff and non-stiff constituent parts. These are then advanced in time using an implicit and explicit scheme, respectively. By design, AIMEX schemes have an arbitrarily large number of stages. This allows for optimization of the explicit part to improve its stability properties, increasing the allowable time step size. Importantly, only two implicit stages are required regardless of the total number of stages, meaning a larger global time step can be taken with significantly fewer implicit stages for a given simulation time. Numerical results demonstrate that AIMEX schemes achieve their designed order of accuracy for linear and non-linear problems involving mesh induced stiffness. Simulations of unsteady flow over an SD7003 airfoil using the compressible Navier-Stokes equations demonstrate that AIMEX schemes can significantly outperform classical explicit Runge-Kutta schemes by over a factor of 20, and conventional IMEX schemes by over a factor of two, with negligible impact on quantitative results. Finally, a demonstration case of Implicit Large Eddy Simulation (ILES) of flow over a stalled NACA0020 airfoil shows the utility of AIMEX schemes for wall-bounded turbulent flows.

A Precise-Integration Time-Domain Formulation Based on Auxiliary Differential Equation for Transient Propagation in Plasma

The electromagnetic wave propagation through plasma medium is one of the most important research fields in computational electromagnetics. A numerical formulation based on both the auxiliary differential equation (ADE) and the precise-integration time-domain (PITD) method for solving the plasma problems is proposed to break through the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the current density J is introduced as the auxiliary variable to deal with the complex permittivity of the plasma which is dependent on the frequency, and the precise integration (PI) technique makes the selectable maximum time-step size become much larger and removes the impact of the time-step size to the numerical dispersion error. Numerical experimentations of the typical plasma problems verify and validate the reliability of the proposed formulation. Through the numerical results, it can be found that the maximum allowable time-step size of the new method is much larger than that of the CFL limit of the FDTD method, and the calculation error of the new method is nearly independent of the time-step size. As a consequence, the execution time is significantly reduced by using a larger time-step size.

Large eddy simulation of turbomachinery flows using a high-order implicit residual smoothing scheme

A recently developed fourth-order accurate implicit residual smoothing scheme (IRS4) is investigated for the large eddy simulation of turbomachinery flows, characterized by moderate to high Reynolds numbers and subject to severe constraints on the maximum allowable time step if an explicit scheme is used. For structured multi-block meshes, the proposed approach leads to the inversion of a scalar pentadiagonal system by mesh direction, which can be done very efficiently. On the other hand, applying IRS4 at each stage of an explicit Runge–Kutta time scheme allows to increase the time step by a factor 5 to 10, leading to substantial savings in terms of overall computational time. With respect to standard second-order fully implicit approaches, the IRS4 does not require approximate linearization and factorization procedures nor inner Newton-Raphson subiterations. As a consequence, it represents a better cost-accuracy compromise for the numerical simulations of turbulent flows where the maximum time step is controlled by the lifetime of the smallest resolved turbulent structures. Numerical results for the well-documented high-pressure VKI LS-89 planar turbine cascade illustrate the potential of IRS4 for significantly reducing the overall cost of turbomachinery large eddy simulations, while preserving an accuracy similar to the explicit solver even for sensitive quantities like the heat transfer coefficient and the turbulent kinetic energy field.

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge–Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge–Kutta methods were established in Isherwood et al. (SIAM J Numer Anal 56(6):3276–3307, 2018), where it was shown that it is possible to define explicit integrating factor Runge–Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge–Kutta method that has non-decreasing abscissas. However, explicit SSP Runge–Kutta methods have an order barrier of $$p=4$$, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge–Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge–Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge–Kutta methods; furthermore, our explicit SSP two-step Runge–Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge–Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge–Kutta methods designed in Isherwood et al. (2018).

L2-stability analysis of IMEX-(σ,μ)DG schemes for linear advection-diffusion equations

A fully discrete L2-stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the (σ,μ)-family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L2-stability for time steps Δt=O(d/a2), where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent (14,94)-recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stability property which is also confirmed by numerical experiments.

Verification of the Time-Split Method for Higher-Order Diffusion in the Spectral Element Method Model on a Cubed-Sphere Grid

This study used the time-split method for higher-order diffusion in a global numerical weather prediction model of the spectral element method on a cubed-sphere grid, and verified its performance compared to those of the explicit and implicit timestep diffusion schemes. Because the spectral element method on a cubed-sphere grid involves intrinsic characteristics of eigenvalues of the Laplacian operator, allowable timestep sizes are more restricted for stable time integration in higher-order diffusion. For example, the 6th-order diffusion significantly limits the timestep size for adequate filtering of small-scale numerical noise during simulation. The time-split scheme can be an alternative to overcome the severe constraints on the timestep size and allow the model a timestep size as large as is potentially acceptable even in higher-order diffusion, while maintaining atmospheric kinetic energy compared to that of the explicit and implicit schemes. During analysis of the amplification factor and numerical simulation of the schemes, it was shown that the time-split scheme has a theoretical response comparable to that of the explicit and implicit schemes. The numerical simulations were conducted using the Korean Integrated Model (KIM) with a 6th-order diffusion operator of three different diffusion schemes at resolutions of 0.25- or 0.125-degree grid spacing. The time-split scheme yields a forecast performance similar to both the explicit and implicit timestep schemes while greatly reducing the computational time.

Multi-rate time integration on overset meshes

Overset meshes are an effective tool for the computational fluid dynamic simulation of problems with complex geometries or multiscale spatio-temporal features. When the maximum allowable timestep on one or more meshes is significantly smaller than on the remaining meshes, standard explicit time integrators impose inefficiencies for time-accurate calculations by requiring that all meshes advance with the smallest timestep. With the targeted use of multi-rate time integrators, separate meshes can be time-marched at independent rates to avoid wasteful computation while maintaining accuracy and stability. This work applies time-explicit multi-rate integrators to the simulation of the compressible Navier-Stokes equations discretized on overset meshes using summation-by-parts (SBP) operators and simultaneous approximation term (SAT) boundary conditions. We introduce a class of multi-rate Adams-Bashforth (MRAB) schemes that offer significant stability improvements and computational efficiencies for SBP-SAT methods. We present numerical results that confirm the efficacy of MRAB integrators, outline a number of implementation challenges, and demonstrate a reduction in computational cost enabled by MRAB. We also investigate the use of our method in the setting of a large-scale distributed-memory parallel implementation where we discuss concerns involving load balancing and communication efficiency.

Towards developing an adaptive time stepping for compressible unsteady flows

PurposeThe purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive time-stepping (ATS)-based local error control instead of CFL-based time-stepping technique. Also, an all-speed flow algorithm is implemented with simple low dissipation AUSM convective scheme, which can be computed without preconditioning which in general destroys the time accuracy.Design/methodology/approachIn transient flow computations, the time-step is generally determined from the CFL condition. In this paper, the authors demonstrate the usefulness of ATS based on local time-stepping previously used extensively in ordinary differential equations (ODE) integration. This method is implemented in an implicit framework to ensure the numerical domain of dependence always contains the physical domain of dependence.FindingsIn this paper, the authors limit their focus to capture the unsteady physics for three cases: Sod’s shock-tube problem, Stokes’ second problem and a circular cylinder. The use of ATS with local truncation error control enables the solver to use the maximum allowable time-step, for the prescribed tolerance of error. The algorithm is also capable of converging very rapidly to the steady state (if there is any) after the initial transient phase. The authors present here only the first-order time-stepping scheme. An algorithmic comparison is made between the proposed adaptive time-stepping method and the commonly used dual time-stepping approach that indicates the former will be more efficient.Originality/valueThe original method of ATS based on local error control is used extensively in ODE integration, whereas, this method is not so popular in the computational fluid dynamics (CFD) community. In this paper, the authors investigate its use in the unsteady CFD computations. The authors hope that it would provide CFD researchers with an algorithm based on an adaptive time-stepping approach for unsteady calculations.

Numerical simulation of plastic deformation and mechanical response of strip rolled aluminium alloys

During the optimal design and simulation of machining or forming processes, detailed simulation of structural response is typically required for use in Finite Element Analysis (FEA). In this study, the bulk temperature and rate dependent resistance to deformation of strip formed aluminium alloys is modelled using the Mechanical Threshold Stress model. The model is characterised to AA5182 alloy data and used in the FEA simulation of a strip rolling process. The effects of element choice, dynamic or quasi-static simulations and the use of particular modelling algorithms on the computational cost and overall accuracy associated with each simulation have to be considered. Given the correctly implemented material tangent, an implicit analysis is illustrated to allow larger stable time-steps. Dynamic and quasi-static solutions are very similar for the simulated process meaning inertia effects are negligible. It is further demonstrated that great care be given to the maximum allowable time step size in order to capture the expected interaction between temporal and spatial discretisation in a fully Lagrangian FEA simulation.

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.