Adaptive Time-stepping Scheme Research Articles

Towards a new modelling of gas flows in a semi-analytical model of galaxy formation and evolution

We present an extended version of the semi-analytical model, GalICS. Like its predecessor, eGalICS applies a post-treatment of the baryonic physics on pre-computed dark-matter merger trees extracted from an N-body simulation. We review all the mechanisms that affect, at any given time, the formation and evolution of a galaxy in its host dark-matter halo. We mainly focus on the gas cycle from the smooth cosmological accretion to feedback processes. To follow this cycle with high accuracy we introduce some novel prescriptions: i) a smooth baryonic accretion with two phases: a cold mode and a hot mode built on the continuous dark-matter accretion. In parallel to this smooth accretion, we implement the standard photoionisation modelling to reduce the input gas flow on the smallest structures. ii) a complete monitoring of the hot gas phase. We compute the evolution of the core density, the mean temperature and the instantaneous escape fraction of the hot atmosphere by considering that the hot gas is in hydrostatic equilibrium in the dark-matter potential well, and by applying a principle of conservation of energy on the treatment of gas accretion, supernovae and super massive black hole feedback iii) a new treatment for disc instabilities based on the formation, the migration and the disruption of giant clumps. The migration of such clumps in gas-rich galaxies allows to form pseudo-bulges. The different processes in the gas cycle act on different time scales, and we thus build an adaptive time-step scheme to solve the evolution equations. The model presented here is compared in detail to the observations of stellar-mass functions, star formation rates, and luminosity functions, in a companion paper.

A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.

Development and Analysis of a Block-Preconditioner for the Phase-Field Crystal Equation

We develop a preconditioner for the linear system arising from a finite element discretization of the phase-field crystal (PFC) equation. The PFC model serves as an atomic description of crystalline materials on diffusive time scales and thus offers the opportunity to study long time behavior of materials with atomic details. This requires adaptive time stepping and efficient time-discretization schemes, for which we use an embedded Rosenbrock scheme. To resolve spatial scales of practical relevance, parallel algorithms are also required, which scale to large numbers of processors. The developed preconditioner provides such a tool. It is based on an approximate factorization of the system matrix and can be implemented efficiently. The preconditioner is analyzed in detail and shown to speed up the computation drastically.

An adaptive time-stepping strategy for the implicit solution of steady transonic flow

This study addresses a novel adaptive time stepping procedure, which leads to selection of larger time steps allowed by the physics of the problem under consideration. Information about the gradients of the flow variables can be regarded as an indicator for determining proper amount of time step, in which the system evolved. In this study, the signals from the pressure sensors, which act according to the pressure gradients, are chosen as a measure to determine the magnitude of the local CFL number. Thus, the aimed methodology for the selection of the local time step with the use of 'pressure sensor' introduces optimal time steps to the implicit solution method by accounting for the pressure gradient in the solution domain, such that sharp pressure gradients encourages small time steps and vice versa. To illustrate the effect of proposed procedure, Newton Krylov (NK), with implicit pseudo time stepping method, has been employed to solve the compressible Euler equations for steady transonic case by turning on the pressure switch. Numerical experiments show that the introduced adaptive time stepping procedure decreases the computation time and the number of iterations, effectively.

Adaptive time-stepping for the strong numerical solution of stochastic differential equations

Models based on stochastic differential equations are of high interest today due to their many important practical applications. Thus the need for efficient and accurate numerical methods to approximate their solution. In this paper, we propose several adaptive time-stepping strategies for the strong numerical solution of stochastic differential equations in Ito form, driven by multiple Wiener processes satisfying the commutativity condition. The adaptive schemes are based on I and PI control, and allow arbitrary values of the stepsize. The explicit Milstein method is applied to approximate the solution of the problem and the adaptive implementations are based on estimates of the local error obtained using Richardson extrapolation. Numerical tests on several models arising in applications show that our adaptive time-stepping schemes perform better than the fixed stepsize alternative and an adaptive Brownian tree time-stepping strategy.

An Adaptive Time-Stepping Scheme with Local Convergence Verification Using Support Vector Machines

An Adaptive Time-Stepping Scheme with Local Convergence Verification Using Support Vector Machines

An adaptive time-stepping strategy for solving the phase field crystal model

In this work, we will propose an adaptive time step method for simulating the dynamics of the phase field crystal (PFC) model. The numerical simulation of the PFC model needs long time to reach ste...

An efficient implicit‐explicit adaptive time stepping scheme for multiple‐time scale problems in shear zone development

Problems associated with shear zone development in the lithosphere involve features of widely different time scales, since the gradual buildup of stress leads to rapid and localized shear instability. These phenomena have a large stiffness in time domain and cannot be solved efficiently by a single time‐integration scheme. This conundrum has forced us to use an adaptive time‐stepping scheme, in particular, the adaptive time‐stepping scheme (ATS) where the former is adopted for stages of quasi‐static deformation and the latter for stages involving short time scale nonlinear feedback. To test the efficiency of this adaptive scheme, we compared it with implicit and explicit schemes for two different cases involving: (1) shear localization around the predefined notched zone and (2) asymmetric shear instability from a sharp elastic heterogeneity. The ATS resulted in a stronger localization of shear zone than the other two schemes. We report that usual implicit time step strategy cannot properly simulate the shear heating due to a large discrepancy between rates of overall deformation and instability propagation around the shear zone. Our comparative study shows that, while the overall patterns of the ATS are similar to those of a single time‐stepping method, a finer temperature profile with greater magnitude can be obtained with the ATS. The ability to model an accurate temperature distribution around the shear zone may have important implications for more precise timing of shear rupturing.

An adaptive time-stepping strategy for solving the phase field crystal model

In this work, we will propose an adaptive time step method for simulating the dynamics of the phase field crystal (PFC) model. The numerical simulation of the PFC model needs long time to reach steady state, and then large time-stepping method is necessary. Unconditionally energy stable schemes are used to solve the PFC model. The time steps are adaptively determined based on the time derivative of the corresponding energy. It is found that the use of the proposed time step adaptivity cannot only resolve the steady state solution, but also the dynamical development of the solution efficiently and accurately. The numerical experiments demonstrate that the CPU time is significantly saved for long time simulations.

Non-iterative adaptive time-stepping scheme with temporal truncation error control for simulating variable-density flow

The automatic non-iterative second-order time-stepping scheme based on the temporal truncation error proposed by Kavetski et al. [Kavetski D, Binning P, Sloan SW. Non-iterative time-stepping schemes with adaptive truncation error control for the solution of Richards equation. Water Resour Res 2002;38(10):1211, http://dx.doi.org/10.1029/2001WR000720.] is implemented into the code of the HydroGeoSphere model. This time-stepping scheme is applied for the first time to the low-Rayleigh-number thermal Elder problem of free convection in porous media [van Reeuwijk M, Mathias SA, Simmons CT, Ward JD. Insights from a pseudospectral approach to the Elder problem. Water Resour Res 2009;45:W04416, http://dx.doi.org/10.1029/2008WR007421.], and to the solutal [Shikaze SG, Sudicky EA, Schwartz FW. Density-dependent solute transport in discretely-fractured geological media: is prediction possible? J Contam Hydrol 1998;34:273–91] problem of free convection in fractured-porous media. Numerical simulations demonstrate that the proposed scheme efficiently limits the temporal truncation error to a user-defined tolerance by controlling the time-step size. The non-iterative second-order time-stepping scheme can be applied to (i) thermal and solutal variable-density flow problems, (ii) linear and non-linear density functions, and (iii) problems including porous and fractured-porous media.

Explicit integration of a thermo-mechanical model for clays

This paper considers the numerical implementation of an existing constitutive model that was developed to simulate the thermo-mechanical behavior of clays. The model, which combines two plastic mechanisms, has been reformulated using generalized stress and strain variables accounting for the presence of temperature as an additional variable.Mixed control conditions are introduced to simulate standard thermo-mechanical laboratory tests. An explicit integration scheme is used with an adaptive time-stepping scheme specifically designed for the considered thermo-mechanical model. This sub-stepping procedure enables the constitutive model to be integrated accurately and efficiently by automatically adjusting the size of sub-steps to maintain the local integration error below a specified tolerance. Special attention is paid to the cases where the two yield surfaces are activated simultaneously. The correction of the drift of the stress point from the yield surface is also accounted for.This thermo-elastoplastic model has been implemented in a constitutive driver aiming at simulating thermo-mechanical tests in the triaxial stress space. The efficiency of this integration algorithm is discussed and the clear advantage of using the adaptive sub-stepping procedure is shown. The constitutive model is finally used to simulate thermo-mechanical tests. The results are compared to available experimental data and show the good capabilities of the constitutive model.

Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts

The article is devoted to the study of higher order time integration methods for multibody systems with unilateral constraints. After a brief presentation of the mathematical modeling of nonsmooth multibody systems, several estimates on the local error of consistency of the Moreau time-stepping scheme are given. Based on these estimates, an attempt at an adaptive time-step strategy is presented with academic examples. Finally, higher order event-capturing methods are designed by coupling implicit Runge–Kutta schemes and Moreauʼs time-stepping scheme.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

Enzo+moray: radiation hydrodynamics adaptive mesh refinement simulations with adaptive ray tracing

We describe a photon-conserving radiative transfer algorithm, using a spatially-adaptive ray tracing scheme, and its parallel implementation into the adaptive mesh refinement (AMR) cosmological hydrodynamics code, Enzo. By coupling the solver with the energy equation and non-equilibrium chemistry network, our radiation hydrodynamics framework can be utilised to study a broad range of astrophysical problems, such as stellar and black hole (BH) feedback. Inaccuracies can arise from large timesteps and poor sampling, therefore we devised an adaptive time-stepping scheme and a fast approximation of the optically-thin radiation field with multiple sources. We test the method with several radiative transfer and radiation hydrodynamics tests that are given in Iliev et al. (2006, 2009). We further test our method with more dynamical situations, for example, the propagation of an ionisation front through a Rayleigh-Taylor instability, time-varying luminosities, and collimated radiation. The test suite also includes an expanding H II region in a magnetised medium, utilising the newly implemented magnetohydrodynamics module in Enzo. This method linearly scales with the number of point sources and number of grid cells. Our implementation is scalable to 512 processors on distributed memory machines and can include radiation pressure and secondary ionisations from X-ray radiation. It is included in the newest public release of Enzo.

Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in ℝ2

We study the numerical solution of semilinear parabolic PDEs on unbounded spatial domains ? in ?2 whose solutions blow up in finite time. Of particular interest are the cases where ?=?2 or ? is a sectorial domain in ?2. We derive the nonlinear absorbing boundary conditions for corresponding, suitably chosen computational domains and then employ a simple adaptive time-stepping scheme to compute the solution of the resulting system of semilinear ODEs. The theoretical results are illustrated by a broad range of numerical examples.

A synergic simulation–optimization approach for analyzing biomolecular dynamics in living organisms

A synergic duo simulation–optimization approach was developed and implemented to study protein–substrate dynamics and binding kinetics in living organisms. The forward problem is a system of several coupled nonlinear partial differential equations which, with a given set of kinetics and diffusion parameters, can provide not only the commonly used bleached area-averaged time series in fluorescence microscopy experiments but more informative full biomolecular/drug space–time series and can be successfully used to study dynamics of both Dirac and Gaussian fluorescence-labeled biomacromolecules in vivo. The incomplete Cholesky preconditioner was coupled with the finite difference discretization scheme and an adaptive time-stepping strategy to solve the forward problem. The proposed approach was validated with analytical as well as reference solutions and used to simulate dynamics of GFP-tagged glucocorticoid receptor (GFP-GR) in mouse cancer cell during a fluorescence recovery after photobleaching experiment. Model analysis indicates that the commonly practiced bleach spot-averaged time series is not an efficient approach to extract physiological information from the fluorescence microscopy protocols. It was recommended that experimental biophysicists should use full space–time series, resulting from experimental protocols, to study dynamics of biomacromolecules and drugs in living organisms. It was also concluded that in parameterization of biological mass transfer processes, setting the norm of the gradient of the penalty function at the solution to zero is not an efficient stopping rule to end the inverse algorithm. Theoreticians should use multi-criteria stopping rules to quantify model parameters by optimization.

Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains

This paper is concerned with the numerical solution of semilinear parabolic PDEs on unbounded spatial domains whose solutions blow up in finite time. The focus of the presentation is on the derivation of the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional computational domains and on a simple but efficient adaptive time-stepping scheme. The theoretical results are illustrated by a broad range of numerical examples, including problems with multiple blow-up points.

Solution of the MoMaS reactive transport benchmark with MIN3P—model formulation and simulation results

This paper summarizes the governing equations as implemented in the MIN3P multicomponent flow and reactive transport code (Mayer et al., Water Resour Res 38:1174, 2002) and introduces the equations in discretized form. Linearization and solution methods are presented including adaptive time stepping and update modification schemes. Code-specific details for the implementation of the GdR MoMaS benchmark simulations (Carrayrou et al., Comput Geosci, 2009) are presented. The standard version of the MIN3P code was used to solve the Easy, Medium, and Hard Test Cases, in one and two spatial dimensions, for both advection- and diffusion-dominated conditions. An analysis of the sensitivity of the solution in relation to spatial and temporal discretization parameters is provided for the Easy Test Case, selected results are presented for the Medium and Hard Test Cases, and the performance of the code as a function of discretization parameters is evaluated for all test cases.

A modified truncation error-based adaptive time-step control strategy for fully- and semi-implicit simulation of isothermal solidification processes

When analyzing the transient characteristics of solidification processes, choosing appropriately-sized time steps is difficult. Accordingly, the current study develops a modified local time truncation error (LTE)-based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the time steps can be adapted in accordance with the local variations in latent heat released during phase change or the effects of pure conduction in a single solid or liquid phase. In the approach presented in this work, the LTE-based time-step evaluation procedure is applied not only after a convergent temperature field is obtained at each time step, but also during the nonlinear iterations performed at each time step whenever a convergence problem is encountered. The computational accuracy and efficiency of the proposed method are demonstrated via the simulation of the one-dimensional and two-dimensional solidification problems and compared with those of other adaptive time step and the uniform time step methods. Furthermore, the performance of these approaches has also been demonstrated using fully-implicit and semi-implicit schemes.

An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: Chemical Langevin equation, a proof of concept

Models involving stochastic differential equations (SDEs) play a prominent role in a wide range of applications where systems are not at the thermodynamic limit, for example, biological population dynamics. Therefore there is a need for numerical schemes that are capable of accurately and efficiently integrating systems of SDEs. In this work we introduce a variable size step algorithm and apply it to systems of stiff SDEs with multiple multiplicative noise. The algorithm is validated using a subclass of SDEs called chemical Langevin equations that appear in the description of dilute chemical kinetics models, with important applications mainly in biology. Three representative examples are used to test and report on the behavior of the proposed scheme. We demonstrate the advantages and disadvantages over fixed time step integration schemes of the proposed method, showing that the adaptive time step method is considerably more stable than fixed step methods with no excessive additional computational overhead.