Adaptive Time Step Research Articles

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.

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

Numerical simulation of three-phase mud invasion using a epidermal effect model and adaptive time-stepping

Numerical simulation of three-phase mud invasion using a epidermal effect model and adaptive time-stepping

Modelling of wall-bounded cavitating flow and spray mixing in multi-component environments using the PC-SAFT equation of state

This work introduces a numerical multiphase model for multi-component mixtures, utilizing tabulated data for physical and transport properties across a spectrum of conditions from near-vacuum pressures to supercritical states. The property data are derived using Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT), vapor-liquid equilibrium (VLE) calculations, entropy scaling methodologies, and Group Contribution (GC) methods. These techniques accurately reflect the thermodynamic behaviors of real fluids, avoiding the empirical estimation of Equation of State (EoS) input parameters. Implemented in OpenFOAM, the fluid dynamics solver is designed to address the three-dimensional Navier-Stokes equations for multi-component mixtures. The methodology integrates operator splitting to manage hyperbolic and parabolic steps distinctively. Hyperbolic terms are solved using the HLLC (Harten-Lax-van Leer-Contact) solver with temporal integration performed via a third-order Strong-Stability-Preserving Runge–Kutta (SSP-RK3) method. Viscous stress tensor contributions in the momentum equation are handled through an implicit velocity correction equation, while parabolic terms in the energy equation are explicitly solved. The simulation efficiency is further enhanced by adaptive Local Time Stepping and the Immersed Boundary (IB) method, which addresses interactions between the fluid and solid boundaries. Turbulence is resolved using the Wall Adaptive Large Eddy (WALE) model. Applied to high-pressure diesel fuel spray injections into non-reacting (nitrogen) gas environments, the model has been validated against Engine Combustion Network (ECN) data for the Spray-C configuration, featuring a fully cavitating multi-hole orifice. Results demonstrate that the model achieves accurate predictions across a broad range of tested conditions without the need for tuning or calibration parameters.

A numerical study of vortex nucleation in 2D rotating Bose–Einstein condensates

This article implements a numerical method for the minimization under constraints of a discrete energy modelling multicomponents rotating Bose–Einstein condensates in the regime of strong confinement and with rotation. Moreover, this method allows to consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). The goal of this paper is to study numerically the structures of the minimizers. In order to do so, we introduce a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

Condensed Galerkin Time-Element for Structural Dynamics with Adaptive Time-Stepping in Maximum Norm

In this paper, the newly-developed condensed Galerkin element of polynomial degree [Formula: see text] for first-order initial-value problems is reformulated for and extended to the structural dynamic equations, yielding a high-performance dynamic element of one-step, unconditionally stable type with the conventional finite element (FE) convergence [Formula: see text] within the element and the extra-superconvergence [Formula: see text] at the element end-nodes. Further, based on the element energy projection (EEP) technique, a superconvergence EEP formula for the condensed element is proposed in the paper and mathematical analysis is presented to prove that the derived EEP solution gains a superconvergence [Formula: see text], which is an order higher than the FE solution of [Formula: see text] for all element degree [Formula: see text] and is hence qualified to serve as a point-wise error estimator. As a result, a simple, efficient, and reliable algorithm with adaptive time-stepping controlled by the maximum norm is proposed. Representative numerical examples are presented to verify the validity of the proposed theorems and to demonstrate the high performance of the proposed element and the associated adaptivity algorithm.

A convection-diffusion-reaction system with discontinuous flux modelling biofilm growth in slow sand filters

A Slow Sand Filter (SSF) consists of biofilm attached to saturated packed sand with supernatant water on top, through which water flows by gravity. SSFs are used in the production of drinking water to improve water quality and remove pathogens, and, as they use biological filtration via the microbes present in the biofilm, they are simple and cheap to construct. A drawback of SSFs is that the difficult sampling and the complex dynamics associated with biofilm growth of microbes pose an obstacle for analysis and predictions. In order to overcome this issue, a mathematical model consisting of a non-linear convection-diffusion-reaction system with discontinuous flux is developed from first principles as a unified framework for the dynamics in time and depth of the biofilm and microbial community, where a variety of ecological models can be included in the reaction terms, which also contain attachment, detachment and transfer processes. The multi-phase approach considers three main physical volumes: a biofilm matrix, the water volume enclosed by it and the flowing suspension through the SSF surrounding the biofilm. The only input data needed are the inflow concentrations and rate, light irradiation and temperature. The model includes a Cahn–Hilliard-type equation for the biofilm velocity in the supernatant water. A numerical scheme that preserves physical properties of the solution is also presented. This is achieved with a time-splitting scheme using an upwind scheme with an adaptive time-step size that ensures positivity of the solution under a CFL condition and a reasonable assumption on the separate numerical solution of the Cahn–Hilliard-type equation. The latter is solved numerically with a semi-implicit finite-difference scheme using a mixed form of the equation. The flexible model framework with its numerical scheme allows for including a variety of ecological models, making the investigation and development of increasingly complex simulations possible. Simulations with different model parameters are shown to be qualitatively consistent with the literature.

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

In this work, we investigate a linear second-order numerical method for the Allen-Cahn equation with general mobility. The proposed scheme is a combination of the two-step first- and second-order backward differentiation formulas for time approximation and the central finite difference for spatial discretization, two additional stabilizing terms are also included. The discrete maximum bound principle of the numerical scheme is rigorously proved under mild constraints on the adjacent time-step ratio and the two stabilization parameters. Furthermore, the error estimates in H1-norm for the case of constant mobility and L∞-norm for the general mobility case, as well as the energy stability for both cases are obtained. Finally, we present extensive numerical experiments to validate the theoretical results, and develop an adaptive time-stepping strategy to demonstrate the performance of the proposed method.

An improved solution of the probability density evolution equation for analyzing stochastic structural responses with adaptive time steps and initial conditions

The probability density evolution method (PDEM) offers an innovative approach for analyzing the stochastic vibration responses of structures. Given the complexities inherent in real-world challenges, the probability density evolution equations (PDEE) often require numerical methods for effective resolution. Consequently, improving computational efficiency and solution accuracy is of paramount importance. Currently, the impulse function discretization method (IFDM) faces several challenges, including limited differentiability in the spatial direction for initial value conditions. Furthermore, the probability density function produced by this method frequently deviates significantly from the true value, undermining the accuracy of the PDEE. Additionally, when addressing vibration responses characterized by rapid evolutionary velocities, the use of a globally fixed uniform time step size (FUTSS) results in a marked decrease in computational efficiency. To tackle these challenges, this study introduces an improved probability density evolution method (IPDEM), which integrates the Gaussian kernel density estimation discretization method (GKDEDM) and an adaptive non-uniform time step size (ANUTSS). The advantages of the IPDEM are illustrated through two numerical simulation examples. The findings indicate that the GKDEDM not only enhances spatial differentiability but also provides a more accurate estimation of the probability density at the initial moment, thereby improving the accuracy of the PDEE. Furthermore, the ANUTSS significantly reduces the number of iterations required while preserving precision, leading to a substantial optimization of computation time.

Numerical method research and characteristics analysis on frozen start-up process of high-temperature heat pipe

Future space exploration technology requires a long-life and reliable power source that is not reliant on solar energy. Space micro-reactors are able to meet this need, with Heat Pipe Cooled Reactors (HPR) emerging as a notable type of space micro-reactor that has attracted widespread attention in recent years. The HPR utilizes high-temperature alkali metal heat pipes for heat transfer, which presents certain complexities due to the solid state of the alkali metals working medium at room temperature. This results in a three-phase transition during the high-temperature heat pipes start-up process, which significantly impacts the heat transfer characteristics and dynamic behavior of the HPR start-up process. Consequently, thorough research is necessary in this area. Numerical simulation is a crucial tool that can effectively analyze, predict, and guide experiments. This article utilizes the Finite Volume Method (FVM) to develop a simulation code for high-temperature heat pipe frozen start-up. Various physical models are integrated to describe different components of the heat pipe: the container is represented by a two-dimensional axisymmetric heat conduction equation, the wick region utilizes a Fixed Grid Method (FGM) to depict the melting process of the medium, and the vapor channel is described through a two-dimensional axisymmetric compressible laminar flow. The wick region and vapor channel are coupled through the evaporation and condensation of the medium. For the vapor channel, numerical methods such as SIMPLE and PISO are used for solving. Adaptive time step and OpenMP acceleration are employed in the code to enhance computational efficiency. Finally, by comparing the calculated results with experimental data, the feasibility and accuracy of the code are assessed, highlighting special phenomena during the start-up process. The findings confirm that the developed code accurately predicts parameter changes during start-up, and can serve as a heat pipe analysis module for multi-physics coupling analysis of HPR.

A hybrid Eulerian-Lagrangian Vlasov method for nonlinear wave-particle interaction in weakly inhomogeneous magnetic field

We present a hybrid Eulerian-Lagrangian (HEL) Vlasov method for nonlinear resonant wave-particle interactions in weakly inhomogeneous magnetic field. The governing Vlasov equation is derived from a recently proposed resonance tracking Hamiltonian theory. It gives the evolution of the distribution function with a scale-separated Hamiltonian that contains the fast-varying coherent wave-particle interaction and slowly-varying motion about the resonance frame of reference. The hybrid scheme solves the fast-varying phase space evolution on Eulerian grid with an adaptive time step and then advances the slowly-varying dynamics by Lagrangian method along the resonance trajectory. We apply the HEL method to study the frequency chirping of whistler-mode chorus wave in the magnetosphere and the self-consistent simulations reproduce the chirping chorus wave and give high-resolution phase space dynamics of energetic particles at low computational cost. The scale-separated HEL approach could provide additional insights of the wave instabilities and wave-particle nonlinear coherence compared to the conventional Vlasov and particle-in-cell methods.

Strong convergence of a class of adaptive numerical methods for SDEs with jumps

We develop adaptive time-stepping strategies for Itô-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs.Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability.In this article, we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case, where the jump and diffusion perturbations are mutually independent, and the jump coefficient satisfies a global Lipschitz condition.

Parameter identification of vibratory conveying systems including statistical part behavior

This work presents a complete workflow for the parameter identification of an MBS model representing a vibratory conveying system. First, the MBS model built within the multibody dynamics tool HOTINT is shown. Essential components, such as contact modeling and adaptive time step control, are explained in detail. The model includes a considerable amount of parameters that need to be identified. Altough, this model is fully deterministic, the complex contact dynamics can cause major effect on results by minor parameter variations. Thus, parameter variations usually show statistical variance, which is characteristic for real conveying processes too. For parameter identification it is important to detect parameters that significantly correlate with results. A method that allows to distinguish significant and non-significant parameters is presented. The application to the multibody system shows that stiffness, restitution and sliding friction of the contact model are the essential parameters. Next, the parameter identification is discussed with particular attention to multimodal distributions, that can occur in conveying systems. Furthermore, the integration of statistics into the deterministic simulation model is discussed. In order to demonstrate the applicability of the proposed methods, a parameter identification is performed based on measurement data of a single part on a real conveying system in the development site of STIWA Automation GmbH. Finally, the validity of the fully parametrized simulation model is shown by investigating geometric adjustments of the conveyor. In order to further verify the identified set of parameters, a conveying process with multiple parts is analyzed and compared in measurement and simulation.

A Consolidated Linearised Progressive Flooding Simulation Method for Onboard Decision Support

In pursuing quick and precise progressive flooding simulations for decision-making support, the linearised method has emerged and undergone refinement in recent years, becoming a reliable tool, especially for onboard decision support. This study consolidates and enhances the modelling approach based on a system of differential-algebraic equations capable of accommodating compartments filled with floodwater. The system can be linearised to permit analytical solutions, facilitating the utilization of larger time increments compared to conventional solvers for differential equations. Performance enhancements are achieved through the implementation of an adaptive time-step mechanism during the integration process. Furthermore, here, a correction coefficient for opening areas is introduced to enable the accurate modelling of free outflow scenarios, thereby mitigating issues associated with the assumption of deeply submerged openings used in governing equations. Experimental validation is conducted to compare the method’s efficacy against recent model-scale tests, specifically emphasising the improvements stemming from the correction for free outflow.

An adaptive numerical method for multi-cellular simulations of tissue development and maintenance

In recent years, multi-cellular models, where cells are represented as individual interacting entities, are becoming ever popular. This has led to a proliferation of novel methods and simulation tools. The first aim of this paper is to review the numerical methods utilised by multi-cellular modelling tools and to demonstrate which numerical methods are appropriate for simulations of tissue and organ development, maintenance, and disease. The second aim is to introduce an adaptive time-stepping algorithm and to demonstrate it’s efficiency and accuracy. We focus on off-lattice, mechanics based, models where cell movement is defined by a series of first order ordinary differential equations, derived by assuming over-damped motion and balancing forces. We see that many numerical methods have been used, ranging from simple Forward Euler approaches through to higher order single-step methods like Runge–Kutta 4 and multi-step methods like Adams–Bashforth 2. Through a series of exemplar multi-cellular simulations, we see that if: care is taken to have events (births deaths and re-meshing/re-arrangements) occur on common time-steps; and boundaries are imposed on all sub-steps of numerical methods or implemented using forces, then all numerical methods can converge with the correct order. We introduce an adaptive time-stepping method and demonstrate that the best compromise between L∞ error and run-time is to use Runge–Kutta 4 with an increased time-step and moderate adaptivity. We see that a judicious choice of numerical method can speed the simulation up by a factor of 10–60 from the Forward Euler methods seen in Osborne et al. (2017), and a further speed up by a factor of 4 can be achieved by using an adaptive time-step.

Towards modelling AR Sco: generalized particle dynamics and strong radiation-reaction regimes

ABSTRACT Numerical simulations of relativistic plasmas have become more feasible, popular, and crucial for various astrophysical sources with the availability of computational resources. The necessity for high-accuracy particle dynamics is especially highlighted in pulsar modelling due to the extreme associated electromagnetic fields and particle Lorentz factors. Including the radiation-reaction force in the particle dynamics adds even more complexity to the problem, but is crucial for such extreme astrophysical sources. We have also realized the need for such modelling concerning magnetic mirroring and particle injection models proposed for AR Sco, the first white dwarf pulsar. This paper demonstrates the benefits of using higher-order explicit numerical integrators with adaptive time-step methods to solve the full particle dynamics with radiation-reaction forces included. We show that for standard test scenarios, namely various combinations of uniform E- and B-fields and a static dipole B-field, the schemes we use are equivalent to and in extreme field cases outperform standard symplectic integrators in accuracy. We show that the higher-order schemes have massive computational time improvements due to the adaptive time-steps we implement, especially in non-uniform field scenarios and included radiation reaction where the particle gyro-radius rapidly changes. When balancing accuracy and computational time, we identified the adaptive Dormand–Prince eighth-order scheme to be ideal for our use cases. The schemes we use maintain accuracy and stability in describing the particle dynamics and we indicate how a charged particle enters radiation-reaction equilibrium and conforms to the analytical Aristotelian Electrodynamics expectations.

Approach for sizing by optimization of an electrical drive considering a multiphysics and multidynamics behaviour

PurposeThis paper aims to describe a modelling and solving methodology of a (static converter–electric motor–control) system for its sizing by optimization, considering the dynamic thermal heating of the machine.Design/methodology/approachThe electrical drive sizing model is composed of two simulators (electrical and thermal) that are co-simulated with a master−slave relationship for the time step management. The computation is stopped according to simulation criteria.FindingsThis paper details a methodology to represent and size an electrical drive using a multiphysics and multidynamics approach. The thermal simulator is the master and calls the electrical system simulator at a fixed exchange time step. The two simulators use a dedicated dynamic time solver with adaptive time step and event management. The simulation automatically stops on pre-established criteria, avoiding useless simulations.Research limitations/implicationsThis paper aims to present a generic methodology for the sizing by optimization of electrical drives with a multiphysics approach, so the precision and computation time highly depend on the modelling method of each components. A genetic multiobjective optimization algorithm is used.Practical implicationsThe methodology can be applied to size electrical drives operating in a thermally limited zone. The power electronics converter and electrical machine can be easily adapted by modifying their sub-model, without impacting the global model and simulation principle.Originality/valueThe approach enables to compute a maximum operating duration before reaching thermal limits and to use it as an optimization constraint. These system considerations allow to over constrain the electrical machine, enabling to size a smaller machine while guaranteeing the same output performances.

Adaptive time-stepping for aggregation-shattering kinetics

We propose an experimental study of adaptive time-stepping methods for efficient modeling of the aggregation-fragmentation kinetics. Precise modeling of this phenomena usually requires utilization of the large systems of nonlinear ordinary differential equations and intensive computations. We study the performance of three explicit Runge–Kutta time-integration methods and provide simulations for two types of problems: finding of equilibrium solutions and simulations for kinetics with periodic solutions. The first class of problems may be analyzed through the relaxation of the solution to the stationary state at large time. In this case, the adaptive time-stepping may help to reach this state using big steps reducing cost of the calculations without loss of accuracy. In the second case, the problem becomes numerically unstable at certain points of the phase space and may require tiny steps making the simulations very time-consuming. Adaptive criteria allows to increase the steps for most of the remaining point and speedup simulations significantly. Мы предлагаем экспериментальное исследование методов интегрирования по времени с адаптивными шагами по времени для эффективного моделирования кинетики агрегации-фрагментации. Точное моделирование этого явления обычно требует использования больших систем нелинейных обыкновенных дифференциальных уравнений и интенсивных вычислений. Мы исследуем производительность трех явных методов Рунге–Кутты и проводим моделирование для двух типов задач: нахождение равновесных решений и моделирование для кинетики с периодическими решениями. Первый класс задач может быть проанализирован посредством релаксации решения к стационарному состоянию на больших временах. В этом случае адаптивные временные шаги могут помочь достичь этого состояния с использованием больших шагов, снижая стоимость вычислений без потери точности. Во втором случае задача оказывается численно неустойчивой в определенных точках фазового пространства и может потребовать крошечных шагов, что делает моделирование с постоянными шагами очень трудоемким. Адаптивные критерии позволяют увеличить шаги для большинства оставшихся точек и значительно ускорить моделирование.

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation

Abstract It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $\mathscr{C}(\mathscr{O})$-norm and $\dot{H}^{\beta }(\mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $\frac{\beta }{2}$ in time and $\frac{\beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $\beta \in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.