Adaptive Time-stepping Method Research Articles

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.

Adaptive conservative time integration for unsteady compressible flow

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

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.

Control-oriented multiphysics model of a lithium-ion battery for thermal runaway estimation under operational and abuse conditions

This study proposes a control-oriented multiphysics model for lithium-ion batteries (LIBs) that can estimate electrochemical-thermal responses in real time under normal operation and abuse conditions. The proposed model integrates the simplified electrochemical model, the thermal resistance network, and the adaptive time-stepping method to ensure computational efficiency without sacrificing the accuracy. Specifically, the diffusion equation of the electrochemical model is simplified by addressing Padé approximation. The thermal resistance network estimates 3D temperature distribution through simple matrix multiplication to account for the entropic, ohmic, and chemical reaction during thermal runaway. The adaptive time-stepping method further secures accurate yet fast explicit calculation. Quantitative experimental validation reveals the high accuracy and robustness, and fast inference time of the proposed model to estimate the electrochemical-thermal responses with the average inference time of 0.0047 s per step. The application of the proposed model on the 26650 LFP cell also demonstrates the versatility on various shapes and types of LIBs. The systematic analysis on the 3D temperature distribution in the LIB of interest not only confirms the effectiveness of the internal temperature monitoring but also ensures the virtual sensing capability. The versatility of the proposed model underscores both design- and control- enabling solutions for battery thermal management.

An adaptive time-stepping Fourier pseudo-spectral method for the Zakharov-Rubenchik equation

An adaptive time-stepping Fourier pseudo-spectral method for the Zakharov-Rubenchik equation

Stochastic Roll Dynamics of Smooth and Impacting Vessels in Random Waves

Ship instability is frequently associated with extreme roll motion. In this paper smooth and vibro-impact dynamics of vessels rolling under random wave excitations including nonlinearities is investigated. The vessel is represented by a single degree of freedom roll model. The random excitation due to waves and ice is modeled by a sum of random pulses with Poisson arrival rate. Stochastic P-bifurcation analysis is carried out by examining the variation with parameter of the joint probability density functions (jpdf) of the response computed numerically by the solution of the corresponding Fokker–Planck–Kolmogorov (FPK) equation. The finite element (FE), finite difference (FD), path integral (PI) and radial basis neural (RBN) network formulation methods are used in the solution of the FPK equation. The D-bifurcation analysis is analyzed by computing the largest Lyapunov exponent. The mean upcrossing rate is computed using Rice’s formula. The random wave excitation is also modeled as the response of a second order linear filter to white noise and the random response of the ship is investigated by solution of the corresponding FPK equation of the four dimensional Markov system by the FD method. Non-smooth coordinate transformations like the Zhuravlev and Ivanov transformations are used converting the discontinuous systems to equivalent smooth systems in the impact dynamics analysis. The adaptive time stepping method (ATSM) approach is used to accurately locate the discontinuity point and a Brownian tree approach is used to direct the integration along the correct Brownian path. Different models for nonlinear damping and the restoring moment are considered and their effects on ship stochastic response such as chaotic motion, unbounded rotational motion, impact modulation motion and ship capsizing are investigated.

Accelerating simulations of Li-ion battery thermal runaway using modified Patankar–Runge–Kutta approach

Reliable and accurate simulations of the thermal runaway of Li-ion batteries are crucial for designing thermal management system. However, the stiff nature of thermal abuse kinetics makes it computationally challenging to obtain numerical solutions. To address this, we employed modified Patankar–Runge–Kutta (MPRK) methods for thermal runaway simulations. The explicit nature of these methods leads to computationally efficient simulations, while their ability to maintain positive physical quantities ensures robust integrations. We first investigated the performance of the second- and third-order MPRK methods, along with the well-known fourth-order Runge–Kutta method, for zero-dimensional thermal runaway simulations. Numerical tests revealed the superiority of the third-order MPRK method in terms of robustness and accuracy. To further improve its accuracy during a rapid temperature rise, we introduced an adaptive time-stepping method. This adaptive third-order MPRK method was then implemented in Ansys Fluent for three-dimensional simulations of thermal runaway propagation. Benchmark tests demonstrated an increase of 10 times in the computational speed compared with Fluent’s built-in solver, which was achieved by allowing significantly larger thermal time steps without compromising accuracy. The robust and efficient computational method proposed in this study will pave the way for future studies on multiphysics simulations of thermal runaway phenomena.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

Time Integration Methods for Nonlinear Dynamic Problems

The accurate simulation and analysis of nonlinear dynamic problems across various scientific disciplines are essential for understanding complex physical phenomena. Time integration methods play a crucial role in numerically solving these problems, providing numerical approximations to the time-evolution of systems governed by ordinary or partial differential equations. This paper presents a comprehensive exploration of time integration methods tailored for addressing nonlinear dynamic problems encountered in diverse fields such as physics, engineering, and computational sciences. The review encompasses a detailed analysis of fundamental time integration techniques, including explicit and implicit schemes, highlighting their respective advantages and limitations in handling nonlinearities. Furthermore, it examines advanced time integration approaches specifically designed to tackle the challenges posed by nonlinear systems. This includes discussions on adaptive time-stepping methods, geometric and symplectic integrators, and other specialized techniques aimed at enhancing accuracy and stability while mitigating computational costs. Through illustrative examples and comparative analyses, this paper evaluates the performance of various time integration methods in addressing nonlinear dynamic problems. It delineates the significance of these methods in accurately capturing system behavior and elucidates their implications for practical applications. Additionally, this review identifies current challenges and outlines prospective directions for further advancements in time integration methodologies tailored to the intricacies of nonlinear dynamic systems. This comprehensive review aims to provide researchers, practitioners, and computational scientists with valuable insights into selecting and implementing appropriate time integration methods for effectively addressing nonlinear dynamic problems across diverse scientific domains. Key Words: Nonlinear Dynamics, Time Integration Techniques, Numerical Simulation, Computational Methods.

Efficient Method for Heat Flux Calculations within Multidisciplinary Analyses of Hypersonic Vehicles

A large amount of heat flux from aerodynamic heating acts on reusable spacecraft; thus, an accurate heat flux prediction around spacecraft reentry is essential for developing a high-performance reusable spacecraft. Although the approximate convective-heating equations can calculate the heat flux with high efficiency and sufficient fidelity, the heat flux should be evaluated over a thousand times for the entire trajectory in multidisciplinary analyses. For these reasons, it is necessary to develop an efficient method for calculating the heat flux for multidisciplinary analysis. In this paper, an efficient method for heat flux calculation that is adoptable by multidisciplinary analyses for hypersonic vehicles, such as spacecraft, is developed. Approximate convective-heating equations were adopted to relieve the computational cost of estimating the heat flux, and an adaptive time step method for heat flux calculations was developed to reduce the number of heat flux calculations required across the entire flight trajectory. A dynamic factor was introduced that adjusts the time step between each instance of the heat flux calculation. Since the time step using this factor could increase under low heat flux conditions, the number of heat flux calculations decreases by approximately one-tenth with over 90% accuracy. Therefore, the efficiency was improved with high accuracy using the adaptively-determined time step according to this dynamic factor.

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

An adaptive time-stepping method for the phase-field molecular beam epitaxial growth model on evolving surfaces

In this paper, a stable and efficient numerical method is presented to simulate the phase-field molecular beam epitaxial growth model on evolving surfaces. The energy dissipation law of the phase-field molecular beam epitaxial growth model on evolving surfaces does not hold for the arbitrary surface velocity which can be considered as a result of external force of the system. However, the stiffness caused by the model’s small interfacial parameter makes conventional numerical approach unstable. Hence, a stabilized semi-implicit method combined with the evolving surface finite element method is proposed and analyzed for the space-time discretization. Moreover, to improve the computational efficiency, an adaptive time-stepping technique which takes account of the energy evolution and surface velocity is also provided. Numerical examples are given to verify the effectiveness of the proposed strategy. Several numerical simulations are performed to explore the beam epitaxial growth on evolving surfaces.

Novel pressure-correction schemes based on scalar auxiliary variable method for the MHD equations

This work introduces a novel schemes that combines scalar auxiliary variable (SAV) and pressure-correction (PC) method to solve the magneto-hydrodynamic (MHD) equations. Prove that the first-order and the second-order PC-SAV scheme are unconditionally energy stable. Moreover, the results obtained by our numerical experiments can well verify the effectiveness of the schemes. Finally, the PC-SAV adaptive time-stepping method is given that improve efficiency.

On approximation classes for adaptive time-stepping finite element methods

Abstract We study approximation classes for adaptive time-stepping finite element methods for time-dependent partial differential equations. We measure the approximation error in $L_2([0,T)\times \varOmega )$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a by-product we define anisotropic Besov spaces for Banach-space-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.

A Hybridizable Discontinuous Galerkin Time-Domain Method With Robin Transmission Condition for Transient Thermal Analysis of 3-D Integrated Circuits

In this article, a discontinuous Galerkin time-domain (DGTD) method hybridized with Robin transmission condition (RTC-DGTD) is proposed for the transient thermal analysis of 3-D integrated circuits (ICs). As a kind of domain decomposition method (DDM), the DGTD method employs a term named numerical flux for the information exchange of solutions between neighboring subdomains. The numerical flux is a function of both temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and heat flux q, and thus for the heat equation, apart from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> , the variable q has to be solved as well. To reach this, the traditional DGTD method decomposes the second-order partial-differential thermal equation into two first-order ones, which results that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and q need to be solved in the whole computational domain, thereby the computational cost is extremely expensive. In this work, to get this trouble-treated, the proposed DGTD method is directly applied to discretize the second-order heat equation. An RTC is established at the interfaces of subdomains, which serves as the second governing equation and meanwhile bridges another connection of solutions in adjacent subdomains. In this way, the new variable q is now only present at the interfaces of subdomains. Consequently, the total number of unknowns is significantly reduced compared with that in the traditional DGTD method. To further speed up the time-marching scheme, an adaptive time-stepping method in terms of the proportional–integral–derivative (PID) control algorithm is employed. Besides, by introducing a restriction operator, the globally coupled matrix equation is torn into a number of small matrix equations pertinent to each subdomain, which is later solved via a finite-element tearing and interconnecting (FETI)-like method. To validate and verify the accuracy and efficiency of the proposed algorithm, several representative examples are studied and compared with COMSOL simulations.

A Fast 3-D finite element modeling algorithm for land transient electromagnetic method with OneAPI acceleration

The land transient electromagnetic (TEM) method is a geophysical prospecting method with a wide range of applications. In this paper, we develop an algorithm that carries out 3-D TEM modeling for loop-source devices. The algorithm is based on the time-domain finite element method with unstructured edge-based meshes, and an unconditional stable adaptive time-stepping method is developed to acquire stable solutions. We verify the algorithm with a 1-D analytical solution. And as case studies, we discuss the TEM responses of both large-loop and small-loop devices in the presence of topography. It is found that the modeling accuracy is more sensitive to space meshing than time discretization. The proposed algorithm is packed as software, TEMF3DT, which is open-source and publicly available together with the above-mentioned sample models. TEMF3DT is designed for PC users, but is also constructed through a higher-level MPI parallelization and a lower-level OpenMP parallelization for potential workstation or cluster uses. Also, TEMF3DT is accelerated by the modern Fortran and Intel OneAPI library to deliver state-of-the-art computational efficiency.

Improved Adaptive Time Step Method for Natural Gas Pipeline Transient Simulation

As the natural gas pipeline network becomes larger and more complicated, a stricter requirement of computation efficiency for the large and complicated network transient simulation should be proposed. The adaptive time step method has been widely used in the transient simulation of natural gas pipeline networks as a significant way to improve computation efficiency. However, the trial calculation process, which is the most time-consuming process in time step adjustment, was used to adjust the time step in these methods, reducing the efficiency of time step adjustment. In order to reduce the number of trial calculations, and improve the calculation efficiency, an improved adaptive time step method is proposed, which proposes the concept of energy number and judges the energy number of the boundary conditions after judging whether the variation of the pipeline state is tolerable. A comparison between the adaptive time step method and the improved adaptive time step method in the restart process of natural gas pipelines and an actual operation of the XB section in China shows the accuracy, effect, and efficiency of the improved adaptive time step method. The results show that with the same accuracy, 27% fewer trial calculation processes and 24.95% fewer time levels are needed in the improved time step method.

On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes.

Numerical analysis of two-phase acidizing in fractured carbonate rocks

A three-dimensional two-phase acidizing model is developed to simulate the acid treatment in fractured carbonate rocks based on a unified pipe-network method. A sequential implicit time technique and an adaptive time step method are used to precisely capture the acidizing process. The Corey model is used to characterize the acid displacement in the water/oil system of the rock matrix, while the Brooks and Corey model is used for the two-phase fluid flow in fractures. The stability and reliability of the current numerical method are evaluated and confirmed by performing numerical verification and convergence analysis with different grid densities. A sensitivity analysis with respect to initial water saturation shows improved acidizing efficiency using the optimized injection rate with a low value of initial water saturation. The effects of fractures, wettability, and porosity heterogeneity on wormhole propagation in fractured carbonate rocks are also investigated. Fractures with dip angles greater than 45° maintain a high acidizing efficiency. More acid is required to abtain acid breakthrough for a fracture-dominant system. However, the injected acid pore volumes to breakthrough (PVBT) decrease for a matrix-dominant system. Higher acidizing efficiency can be achieved if the rock matrix and fractures are water-wet for a fractured carbonate rock. The PVBT in the matrix-dominant system drops markedly when porosity heterogeneity magnitude (dφm) increases from 0.025 to 0.125, and grows when dφm increases from 0.125 to 0.175 due to the generation of multiple wormhole branches. These results provide insights for field-acidizing treatment: Maintaining a high level of initial oil saturation can help increase the acidizing efficiency. A water-wet environment is preferred for acidizing fractured carbonate rocks. Higher pore volumes of acid are required before acid breakthrough in a fracture-dominant system.

Peridynamic Simulation of a Mixed-Mode Fracture Experiment in PMMA Utilizing an Adaptive-Time Stepping for an Explicit Solver

In this paper, a benchmark analysis of a peridynamic correspondence energy-based damage model is presented. The benchmark is an experimental setup of a Polymethyl methacrylate (PMMA) plate with a hole. The plate has a minotch and is subject to a compressive load. With increasing loads, a crack initiates at the tip of the notch and continuously grows. The benchmark is modeled utilizing the peridynamic correspondence formulation as a two-dimensional problem. To reduce numerical issues due to bond failure, an adaptive time-stepping method for a Verlet time integration schema is proposed. The method limits the maximum number of broken bonds per material point by adapting the time-step size. This allows the correspondence formulation to be significantly more stable. The benchmark involves a sensitivity analysis based on the Morris method, which is performed in this context. As a result, uncertainties and the impact of geometrical, numerical and material parameters are evaluated and discussed.