Time Step Size Research Articles

An iterative dynamic chemical stiffness removal method for reacting flow simulations

An iterative dynamic chemical stiffness removal method (IDCSR) based on quasi-steady-state approximation (QSSA) is proposed. The IDCSR method is built on a previously developed non-iterative method which has proved to work well for small timestep sizes. A novel iterative procedure is designed in IDCSR to enable explicit time integration of stiff chemistry at relatively large timestep sizes relevant to practical reacting flow simulations. The effectiveness of the iterative procedure is first demonstrated with a toy problem and homogeneous auto-ignition with fixed integration step sizes, showing that larger timestep sizes can be allowed for explicit time integration using IDCSR compared with the previous non-iterative method. IDCSR is then compared with existing explicit chemistry solvers for simulations of homogeneous auto-ignition and shows similar or lower computational cost but significantly higher accuracy across a wide range of timestep sizes. IDCSR is further combined with an automatic adaptive time-stepping scheme for simulations of 0-D homogeneous auto-ignition and a 2-D laminar lifted n-dodecane jet flame. For the 0-D auto-ignition simulations, IDCSR is shown to reduce both the error (by 43%–90%) and computational cost (by 6–15 times) compared with existing explicit solvers, while achieving speed-up factors of up to 400 compared with VODE for a wide range of timestep sizes and reaction mechanisms. For the 2-D jet flame simulations, speed-up factors of 15 and 31 for chemistry integration, and 5 and 9 for overall simulation, are achieved by IDCSR compared with CVODE with and without analytic Jacobian, respectively.

Two‐Grid Finite Element Method for the Time‐Fractional Allen–Cahn Equation With the Logarithmic Potential

ABSTRACTIn this paper, we propose a two‐grid finite element method for solving the time‐fractional Allen–Cahn equation with the logarithmic potential. Firstly, with the L1 method to approximate Caputo fractional derivative, we solve the fully discrete time‐fractional Allen–Cahn equation on a coarse grid with mesh size and time step size . Then, we solve the linearized system with the nonlinear term replaced by the value of the first step on a fine grid with mesh size and the same time step size . We obtain the energy stability of the two‐grid finite element method and the optimal order of convergence of the two‐grid finite element method in the L2 norm when the mesh size satisfies . The theoretical results are confirmed by arithmetic examples, which indicate that the two‐grid finite element method can keep the same convergence rate and save the CPU time.

Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U-net backbone

Prolonged contact between a corrosive liquid and metal alloys can cause progressive dealloying. For one such process as liquid-metal dealloying (LMD), phase field models have been developed to understand the mechanisms leading to complex morphologies. However, the LMD governing equations in these models often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. In particular, numerical stiffness in the PDEs requires an extremely refined time step size (on the order of 10−12s or smaller). This computational bottleneck is especially problematic when running LMD simulation until a late time horizon is required. This motivates the development of surrogate models capable of leaping forward in time, by skipping several consecutive time steps at-once. In this paper, we propose a U-shaped adaptive Fourier neural operator (U-AFNO), a machine learning (ML) based model inspired by recent advances in neural operator learning. U-AFNO employs U-Nets for extracting and reconstructing local features within the physical fields, and passes the latent space through a vision transformer (ViT) implemented in the Fourier space (AFNO). We use U-AFNOs to learn the dynamics of mapping the field at a current time step into a later time step. We also identify global quantities of interest (QoI) describing the corrosion process (e.g., the deformation of the liquid-metal interface, lost metal, etc.) and show that our proposed U-AFNO model is able to accurately predict the field dynamics, in spite of the chaotic nature of LMD. Most notably, our model reproduces the key microstructure statistics and QoIs with a level of accuracy on par with the high-fidelity numerical solver, while achieving a significant 11, 200 × speed-up on a high-resolution grid when comparing the computational expense per time step. Finally, we also investigate the opportunity of using hybrid simulations, in which we alternate forward leaps in time using the U-AFNO with high-fidelity time stepping. We demonstrate that while advantageous for some surrogate model design choices, our proposed U-AFNO model in fully auto-regressive settings consistently outperforms hybrid schemes.

Performance of the fractional step method with various temporal discretization and adaptive time step size schemes for pulsating ventilation

Performance of the fractional step method with various temporal discretization and adaptive time step size schemes for pulsating ventilation

Step-by-step enhancement of a graph neural network-based surrogate model for Lagrangian fluid simulations with flexible time step sizes

Step-by-step enhancement of a graph neural network-based surrogate model for Lagrangian fluid simulations with flexible time step sizes

Robust Error Estimates for Second‐Order Stabilization Finite Element Method for Navier–Stokes Equations With Small Viscosity and Nonsmooth Initial Data

ABSTRACTIn this article, we study the fully discrete finite element approximation to the two‐dimensional Navier–Stokes equations with a small viscosity coefficient and initial data by using semi‐implicit two‐step backward differentiation formulae (BDF2) method with variable step‐sizes and grad‐div stabilization. The variable step sizes allow us to take special viscosity‐dependent, locally refined time step sizes and thus prove that the variable step‐size BDF2 method can achieve second‐order convergence in time. By adding the grad‐div stabilization, we obtain a robust, fully discrete error bound in which the constants reduce dependence on inverse powers of the viscosity. Numerical results illustrate the effectiveness of the proposed method for such types of Navier–Stokes equations.

Assessment of computational fluid dynamics simulation approach for the aerodynamic performance of small unmanned aerial vehicle propeller

Abstract This study explores the complexities of UAV propeller aerodynamics by examining the influence of various parameters, including mesh resolutions, timestep sizes, iterations per timestep, and different Reynolds-Averaged Navier–Stokes (RANS) turbulence models, using Delayed Detached Eddy Simulation (DDES). Through extensive computational fluid dynamics (CFD) simulations, the aerodynamic performance of the propeller is analyzed under different conditions. Mesh resolutions from coarse to fine are evaluated, with Grid Convergence Index (GCI) and Richardson Extrapolation applied to ensure mesh-independent results. Timestep sizes ranging from 0.5° to 10° are also analyzed, revealing a trade-off between accuracy and computational efficiency. Furthermore, the study assesses the impact of different numbers of iterations per timestep and compares three distinct RANS turbulence models: k-ω SST, Spalart-Allmaras, and Realizable k-ε. The results show that finer mesh resolutions and smaller timesteps enhance accuracy in thrust and torque predictions. The k-ω SST model demonstrates the best balance between thrust generation and torque minimization, while the Spalart-Allmaras model underperforms in both categories. In addition, fewer iterations per timestep were found to be sufficient for convergence, reducing computational cost. Overall, this study provides a comprehensive evaluation of UAV propeller performance using CFD simulations.

A Semi-Analytical Solution to a Generalized Nonlinear Van Der Pol Equation in Plasma By MsDTM

This article examines the effectiveness of the multistage differential transform method (MsDTM) in solving equations with a very strong nonlinear term. It introduces MsDTM as a method for solving the generalized nonlinear Van der Pol equation, which features strong nonlinearity. The generalized nonlinear Van der Pol equation arises in plasma and describes the propagation of various nonlinear phenomena, such as wave propagation in astrophysical plasma. MsDTM demonstrates greater accuracy compared to other analytical and numerical methods, such as the 4th-order Runge-Kutta Method (4thRKM), due to its ability to enhance accuracy through two factors: the number of iterations and the time step size. Most numerical methods rely solely on reducing the time step size to improve accuracy, but for some types of equations, this requires an impractically small time step size, causing the method to fail. In contrast, MsDTM offers an additional means of improving accuracy by increasing the number of iterations. The paper successfully applies MsDTM to solve the Van der Pol equation and presents the results, demonstrating that the method is highly effective for equations with very strong nonlinearity.

Investigation of Tool Cooling and Heat Transfer Using Computational Fluid Dynamics Simulations

Thermal errors remain one of the biggest challenges for the precision of cutting machine tools. Aside from optimizations in the machine tool design and behaviour, optimal cutting process parameters and targeted usage of cutting fluid or alternative methods of tool cooling are required for improved process efficiency with minimal energy demand and maximal tool life. A simulation-based study is presented which compares both different methods of tool cooling, specifically air cooling, flooded cooling and minimum quantity lubrication and also different simulation methods and models. Using a case study, which models an existing thermal test stand comprised of motor spindle, tool holder, tool and coolant nozzle, different cooling scenarios were tested and compared. The simulations were performed in ANSYS CFX. Comparisons were made between simulations with and without buoyancy, with and without tool rotation, transient and steady-state, with laminar flow and with different turbulence models and between the different cooling scenarios. Some insight on different time step sizes and the resulting increase in simulation time and precision was also gained. These results will make future studies on the thermal behaviour of both tool and cutting process easier by showing suitable simulation techniques and viable model simplifications.

Extended Fourier Neural Operators to learn stiff chemical kinetics under unseen conditions

The solution of stiff chemical kinetics is recognized as the computational bottleneck for direct simulations of reacting flows. In this study, we extend the concept of Fourier Neural Operator (FNO) to learn stiff chemical kinetics. Specifically, element and mass conservation are introduced as the physical constraints in the extended FNO (EFNO). In addition, the training data are transformed by a Box–Cox strategy to rectify the skewed distribution of the species in the stiff problems. Finally, balanced loss functions are formulated to address the unbalanced sampling data points in complex reacting flow problems. The EFNO model is leveraged to forecast the temporal evolution of chemical species, utilizing an iterative approach wherein the prediction outcome from the previous time step is employed as a new input for subsequent time step prediction. The results in this work demonstrate the significant use of an EFNO approach to solving stiff chemical dynamics in reacting flow simulations, with a time step size comparable to the typical flow time step size. Its prediction accuracy and generalization ability are evaluated by comparing with the original FNO, Deep Nueral Network (DNN) and DeepONet models, in solving toy problems, zero-dimensional hydrogen autoignition, and a three-dimensional hydrogen/ammonia turbulent jet flame. The EFNO is shown to be highly accurate. More importantly, compared with other deep learning models, it can be generalized to stiff chemical kinetic states under unseen conditions, which the model has never trained for. The great performance of EFNO in terms of accuracy and generalization ability suggests that EFNO is a promising solution algorithm for stiff chemical kinetics problems in reacting flows.Novelty and Significance Statement: The novelty of this work lies in the newly developed extended Fourier neural operators (EFNO) to learn stiff chemical kinetics. Specifically, we for the first time evaluated and tested the performance of Fourier neural operators in solving stiff chemical kinetics. More importantly, we extended the original Fourier neural operators to accurately solve for stiff chemical kinetics problems under unseen conditions, which was a very challenging problem for deep learning methods in the literature. Our results demonstrated that the EFNO model solves chemical kinetics in both simple 0D autoignition and complex 3D turbulent jet flames with great accuracy and generalization ability, even for conditions which the training dataset has never encompassed. This work is significant because it developed a neural operator-based algorithm that can significantly accelerate the stiff chemical kinetic solution process in reacting flow simulations with great accuracy even for unseen initial conditions.

High-order numerical schemes based on B-spline for solving a time-fractional Fokker–Planck equation

The authors of Jiang (2014), Vong and Wang (2014) and Roul et al. (2022) proposed lower-orders computational techniques for solving a time-fractional Fokker–Planck (TFFP) equation. This paper deals with the design of two high-order computational schemes for the TFFP equation. The first scheme is based on a combination of L2−1σ scheme and standard quintic B-spline collocation method, while the second one is based on a combination of L2−1σ scheme and a new technique, namely improvised quintic B-spline collocation method. Convergence of the suggested method is analyzed. An illustrative example is provided to demonstrate the applicability and efficiency of the proposed method. The convergence orders of first and second methods are O(Δt2+Δx4), O(Δt2+Δx6), respectively, where Δt and Δx are the step-sizes in time and space domain, respectively. We compare the computed results with those obtained by the finite difference method (FDM), compact FDM and quartic B-spline collocation method to justify the advantage of proposed schemes.

Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

In this paper, we introduce a novel approach for formulating phase field models with constraints. The main idea is to introduce auxiliary variables that regularize and gradually dissipate constraint deviations of the phase variables, which we name the auxiliary relaxation method. It integrates seamlessly with the energy variational framework to ensure thermodynamic consistency in the resulting phase field models. Unlike traditional penalty methods, which introduce high stiffness due to large penalty parameters to enforce constraints in phase field models, our approach reduces system stiffness, allowing larger time step sizes when solving phase field models with constraints numerically, thus improving numerical accuracy and efficiency. We demonstrate the effectiveness and robustness of the proposed auxiliary relaxation method by applying it across several scenarios to derive thermodynamically consistent phase field models with constraints. Furthermore, we introduce a general second-order implicit-explicit Crank-Nicolson scheme, combining the relaxed scalar auxiliary variable method with a stabilization technique to solve these models. Through extensive numerical tests, we validate the capability of our modeling and numerical framework to reliably simulate complex dynamics governed by phase field equations with constraints.

Improving the Resolution and Computational Efficiency of Depletion Simulations on a TRISO-Fueled Microreactor

Modeling individual TRi-structural ISOtropic (TRISO) particles is possible using constructive solid geometry in Monte Carlo neutron transport codes (such as MCNP); however, the billions of surfaces required incurs considerable computational expense. Depletion simulations, in conjunction with Monte Carlo transport, are also inherently slow, requiring many individual criticality calculations to approximate time-dependent behavior. Additionally, choosing acceptable time step sizes, spatial zoning, and other depletion parameters involves prior knowledge and/or iteration by the user. The slow and complex nature of depletion calculations greatly increases the cost and duration of the design iteration process. The importance of tracked isotopes, spatial zones, and time stepping during depletion simulations is evaluated globally using neutron multiplication factor, total isotope inventory, and burnup. Different depletion schemes are compared using the TRISO-fueled Snowflake microreactor. Woodcock Delta Tracking and a Reactivity-equivalent Physical Transformation (RPT) are investigated as potential approaches to increase the speed of Monte Carlo neutron transport calculations involving TRISO fuel. The Monte Carlo N-Particle® Code (MCNP®) Version 6.3 is used to obtain the transport solution, and depletion calculations are performed using its internally coupled version of CINDER90. A prototype MCNP delta tracking module under development at Los Alamos National Laboratory is also used. Although a single spatial depletion region is sometimes acceptable, large errors in core lifetime and total 238Pu, 241Pu, and 242Pu mass are possible. The prototype delta tracking module significantly speeds up criticality calculations; however, depletion calculations are not always faster than those performed with surface tracking. The RPT method is effective at reducing CPU times in both criticality and depletion calculations, without significantly impacting neutron multiplication factor and total isotope inventory. The necessary number of time steps to converge total isotope inventory was also explored; the required number and spacing varied greatly depending on the objective of the depletion simulation.

Higher-order iterative decoupling for poroelasticity

For the iterative decoupling of elliptic–parabolic problems such as poroelasticity, we introduce time discretization schemes up to order five based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

Superconvergence analysis of a low order expanded conforming mixed finite element method for the extended Fisher–Kolmogorov equation

This paper is devoted to the superconvergence analysis of a low order expanded conforming mixed finite element method(MFEM) for the extended Fisher–Kolmogorov equation. Both semi-discrete and linearized backward Euler fully-discrete schemes are analyzed with bilinear and Nédéleć elements. The present work has two main contributions: One is that some a priori bounds of the approximations are derived, which lead to the unique solvability of the semi-discrete scheme and make the nonlinear term can be dealt with efficiently. By such a way, the superclose properties and superconvergence results can be derived through the high accuracy results of the above two elements and the interpolated postprocessing technique, respectively. The other one is that a novel splitting technique is utilized to deal with the nonlinear term, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size. Besides, the combination technique of interpolation and projection operators also plays an important role in the superclose and superconvergence analysis. Finally, some numerical results are provided to show the validity of the theoretical analysis.

Two-grid finite element methods for space-fractional nonlinear Schrödinger equations

A two-grid finite element method is developed for solving space-fractional nonlinear Schrödinger equations. The finite element solution in L∞-norm is proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the Lp-norm are proved without any time-step size conditions. Finally, the theoretical results are verified by numerical experiments.

Stability and robustness of time-discretization schemes for the Allen-Cahn equation via bifurcation and perturbation analysis

The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently utilized time-discretization numerical schemes for solving the Allen-Cahn equation, with focuses on the Backward Euler, Crank-Nicolson (CN), convex splitting of modified CN, and Diagonally Implicit Runge-Kutta (DIRK) methods. Our stability analysis reveals that the Convex Splitting of the Modified CN scheme exhibits unconditional stability, allowing greater flexibility in time step size selection, while the other schemes are conditionally stable. Additionally, our robustness analysis highlights that the Backward Euler method converges to correct physical solutions regardless of initial conditions. In contrast, all other methods studied in this work show sensitivity to initial conditions and may converge to incorrect physical solutions if the initial conditions are not carefully chosen. This study introduces a comprehensive approach to assessing stability and robustness in numerical methods for solving the Allen-Cahn equation, providing a new perspective for evaluating numerical techniques for general nonlinear differential equations.

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.