Optimal Time Step Research Articles

Effect of a modified sinusoidal forcing on spiral wave in a simulated reaction-diffusion system.

Spiral waves are often found in excitable media. In the hearts, they are abnormal forms of action potential propagation. Under an external forcing, the spiral waves drift and are subsequently terminated at the boundary. Spiral waves can be studied in simulations using a discrete reaction-diffusion system; thereby the time step must not exceed a numerical stability limit (ts). In this article, we present the dynamics of spiral waves in a simulated system under an external forcing as a modified sinusoidal function of time. The spiral waves are forced to drift along a straight line with a velocity and an angle depending on the time step. An optimization study provides the optimal time step of 0.2ts, where further reductions of the time step do not alter the drifting of the spiral waves.

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier–Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Although the resulting adaptive staggered grids are more complex than classical uniform Cartesian meshes, the numerical scheme can still be written in a rather compact form. Thanks to the use of a tensor-product formulation for the definition of the nodal basis in the d-dimensional space (d=2,3), all the discrete operators can be efficiently written as a combination of linear one-dimensional operators acting in the d space directions separately.Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier–Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. In order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive terms in the momentum equation. A particular feature of our staggered DG approach is that the viscous stress tensor is discretized on the dual mesh. This corresponds to the use of a lifting operator, but on the staggered grid. Both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fast convergence is observed. Moreover, it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner. Due to the explicit discretization of the nonlinear convective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicit DG methods based on the fluid velocity. Moreover, by using a suitable local time stepping technique based on the ADER-DG framework, each element can use its optimal local time step also within the AMR context.The new space-time adaptive staggered DG scheme has been thoroughly verified for polynomial degrees up to N=9 for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. The high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution.To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier–Stokes equations on space-time adaptive meshes in two and three space dimensions.

Exploration of optimal time steps for daily precipitation bias correction: a case study using a single grid of RCM on the River Exe in southwest England

ABSTRACTBias correction is a necessary post-processing procedure in order to use regional climate model (RCM)-simulated local climate variables as the input data for hydrological models due to systematic errors of RCMs. Most of the present bias-correction methods adjust statistical properties between observed and simulated data based on a predefined duration (e.g. a month or a season). However, there is a lack of analysis of the optimal period for bias correction. This study attempted to address the question whether there is an optimal number for bias-correction groups (i.e. optimal bias-correction period). To explore this we used a catchment in southwest England with the regional climate model HadRM3 precipitation data. The proposed methodology used only one grid of RCM in the Exe catchment, one emissions scenario (A1B) and one member (Q0) among 11 members of HadRM3. We tried 13 different bias-correction periods from 3-day to 360-day (i.e. the whole of one year) correction using the quantile mapping method. After the bias correction a low pass filter was used to remove the high frequencies (i.e. noise) followed by estimating Akaike’s information criterion. For the case study catchment with the regional climate model HadRM3 precipitation, the results showed that a bias-correction period of about 8 days is the best. We hope this preliminary study on the optimum number bias-correction period for daily RCM precipitation will stimulate more research to improve the methodology with different climatic conditions. Future efforts on several unsolved problems have been suggested, such as how strong the filter should be and the impact of the number of bias correction groups on river flow simulations.Editor M.C. Acreman Associate editor S. Kanae

Efficient photoheating algorithms in time-dependent photoionization simulations

We present an extension to the time-dependent photo-ionization code C$^2$-Ray to calculate photo-heating in an efficient and accurate way. In C$^2$-Ray, the thermal calculation demands relatively small time-steps for accurate results. We describe two novel methods to reduce the computational cost associated with small time-steps, namely, an adaptive time-step algorithm and an asynchronous evolution approach. The adaptive time-step algorithm determines an optimal time-step for the next computational step. It uses a fast ray-tracing scheme to quickly locate the relevant cells for this determination and only use these cells for the calculation of the time-step. Asynchronous evolution allows different cells to evolve with different time-steps. The asynchronized clocks of the cells are synchronized at the times where outputs are produced. By only evolving cells which may require short time-steps with these short time-steps instead of imposing them to the whole grid, the computational cost of the calculation can be substantially reduced. We show that our methods work well for several cosmologically relevant test problems and validate our results by comparing to the results of another time-dependent photo-ionization code.

An optimal time-stepping algorithm for unsteady advection–diffusion problems

We consider the numerical solution of a linear time dependent advection–diffusion problem by an implicit two-weight, three-point finite difference scheme. We extend the scheme proposed by Chadha and Madden (2011), to incorporate an optimal time step selection algorithm for the method. The optimal values of the weights involved in the scheme have been obtained by using the notion of an equivalent differential equation (Warming and Hyett, 1974), and, by enforcing certain constraints due to von Neumann stability and a discrete maximum principle, we obtain sharp bounds for optimal time-stepping in terms of mesh width and problem data. Furthermore, we formulate the time step selection as a root-finding problem, and use this to show the optimal time step exists and is unique. The resulting method, based on optimal values of weights and optimal time-stepping, is of fifth-order in space, and third-order in time. The results of extensive numerical experiments are presented for several classic test problems for various values of the advection and diffusion parameters. These show that our proposed method is superior (quantitatively and qualitatively) to other schemes that can be represented within the framework of two-weighted schemes. We also present comparisons with other high-order methods, and an application of the method to a problem with non-smooth data.

Comparative CFD analysis of Vertical Axis Wind Turbine in upright and tilted configuration

There is a demand of analyzing tilted Vertical Axis Wind Turbine (VAWT) for the emerging applications like VAWTs operating on high rise buildings, Floating Axis Wind Turbine (FAWT) functioning in deep sea, and so on. For this purpose numerical validation of an existing experimental work of Vertical Axis Wind Turbine (VAWT) in upright and tilted conditions has been carried out. The numerical validation is accomplished by means of Computational Fluid Dynamics (CFD) analysis by solving Unsteady Reynolds Averaged Navier–Stokes (URANS) equation. A rigorous parametric study of choosing a right turbulence model for solving URANS equation, mesh dependency analysis and optimum time step is comprehended. After choosing the right parameters, validation is performed. Following that the flow fields of upright and tilted conditions are visualized. The possible reasons of getting higher power coefficient in tilted condition have been discussed. While carrying out the study, it is observed that in case of tilted configuration the wake stream shifts downward. This feature of VAWT in tilted condition could lead into effective sea surface utilization in floating offshore wind farms.

A discussion about optimum time step size and maximum simulation time in EMTP-based programs

Abstract This paper presents a discussion about the optimum time step size and maximum simulation time in EMTP-based programs. This is of particular importance for new users of EMTP-based programs, since the user is responsible for setting up these parameters before running a simulation case. The selection of the time step size affects the precision of the simulation. The time step size depends on the maximum frequency expected in the phenomena, which is normally unknown, a priori. The calculation of eigenvalues from the state space matrix obtained from the EMTP conductance matrix and history terms algorithm formulation allows the exact ideal values for the optimal time step size and a maximum simulation time. In comparison, a general and robust algorithm is also presented here based on all the input data given for the circuit under simulation. The proposed calculation process is based on single- or multi-phase uncoupled or coupled circuits, with lumped or distributed parameters. Simulations are given demonstrating the effectiveness of the proposed rules.

Optimal States Estimation of an LTI System using the Unbiased FIR Filter

The unbiased linear finite impulse response (FIR) filter and the two-state Kalman filter are investigated in the optimal estimation of the two state (position and velocity) in a linear time invariant (LTI) systems in presence of additive white Gaussian noise (AWGN). In opposite to the Kalman filter, the unbiased linear FIR filter dońt need previous knowledge about noise process, this algorithm only needs two specific parameters: optimal time step and optimal number of the points in the average. We show that both algorithms produce a similar lower mean square error (MSE).

Optimal time step for incompressible SPH

A classical incompressible algorithm for Smoothed Particle Hydrodynamics (ISPH) is analyzed in terms of critical time step for numerical stability. For this purpose, a theoretical linear stability analysis is conducted for unbounded homogeneous flows, leading to an analytical formula for the maximum CFL (Courant–Friedrichs–Lewy) number as a function of the Fourier number. This gives the maximum time step as a function of the fluid viscosity, the flow velocity scale and the SPH discretization size (kernel standard deviation). Importantly, the maximum CFL number at large Reynolds number appears twice smaller than with the traditional Weakly Compressible (WCSPH) approach. As a consequence, the optimal time step for ISPH is only five times larger than with WCSPH. The theory agrees very well with numerical data for two usual kernels in a 2-D periodic flow. On the other hand, numerical experiments in a plane Poiseuille flow show that the theory overestimates the maximum allowed time step for small Reynolds numbers.

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.

Ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

We present IAS15, a 15th-order integrator to simulate gravitational dynamics. The integrator is based on a Gau\ss-Radau quadrature and can handle conservative as well as non-conservative forces. We develop a step-size control that can automatically choose an optimal timestep. The algorithm can handle close encounters and high-eccentricity orbits. The systematic errors are kept well below machine precision and long-term orbit integrations over $10^9$ orbits show that IAS15 is optimal in the sense that it follows Brouwer's law, i.e. the energy error behaves like a random walk. Our tests show that IAS15 is superior to a mixed-variable symplectic integrator (MVS) and other popular integrators, including high-order ones, in both speed and accuracy. In fact, IAS15 preserves the symplecticity of Hamiltonian systems better than the commonly-used nominally symplectic integrators to which we compared it. We provide an open-source implementation of IAS15. The package comes with several easy-to-extend examples involving resonant planetary systems, Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment, and generic damping functions that can, among other things, be used to simulate planet-disc interactions. Other non-conservative forces can be added easily.

An adaptive procedure for the numerical parameters of a particle simulation

In this article, a computational procedure that automatically determines the optimum time step, cell weight and species weights for steady-state multi-species DSMC (direct simulation Monte Carlo) simulations is presented. The time step is required to satisfy the basic requirements of the DSMC method while the weight and relative weights fields are chosen so as to obtain a user-specified average number of particles in all cells of the domain. The procedure allows the conduct of efficient DSMC simulations with minimal user input and is integrable into existing DSMC codes. The adaptive method is used to simulate a test case consisting of two counterflowing jets at a Knudsen number of 0.015. Large accuracy gains for sampled number densities and velocities over a standard simulation approach for the same number of particles are observed.

Stability of an overland flow scheme in the framework of a fully coupled eco-hydrological model based on the Macroscopic Cellular Automata approach

Cellular Automata are often used for modeling the evolution in time of environmental systems mainly because they are directly compatible with parallel programming. Nevertheless, defining the optimal time step criterion for integrating forward in time numerical processes can further enhance model computational efficiency. To this aim, a numerical stability analysis of an original overland flow model, within the framework of a fully coupled eco-hydrological system based on the Macroscopic Cellular Automata paradigm, is performed. According to the other modules of the system describing soil water flow, soil-surface-atmosphere fluxes and vegetation dynamics, overland flow model equations were derived through a direct discrete formulation (i.e. no differential equations were discretized), adopting the diffusion wave model as an approximation of the full De Saint Venant equations and including the capability of accounting for specific processes, such as the increasing roughness effects due to vegetation growth or surface-soil water exchanges. Suitable formulations of robust tools usually applied in the stability analyses, such as Courant–Friedrichs–Lewy and von Neumann conditions, were initially derived for the CA-based overland flow model. Afterwards, the theoretical stability conditions were compared to experimental time step constraints through several numerical simulations of a 5-h rain event. Specifically, adopting a constant (i.e. not adaptive) time step for simulations, and discretizing head losses in a way that increases model stability, experimental upper limits preventing numerical instability were found for 13 test cases with different slopes, precipitation intensities, vegetation densities and depths of surface depressions. Even though von Neumann condition and experimental values were well positively correlated, the latter were almost always sensibly lower, excluding cases when free surface gradients tended to zero. Therefore, based on the original method, two alternative criteria were developed. Numerical tests showed that the joint use of these criteria greatly helps in finding the optimal time steps for convergent and stable simulations of the overland flow model.

Determination of Inner and Outer Bounds of Reachable Sets Through Subpavings

The computation of the reachable set of states of a given dynamic system is an important step to verify its safety during operation. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function, and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets of low dimensional systems. The main advantage of this method is that, compared to guaranteed integration, it allows to control the amount of over-estimation at the cost of increased computational effort. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by another guaranteed integration method. The methods developed were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step worse results are obtained.

OPTIMIZATION OF TIME STEP AND CFD STUDY OF COMBUSTION IN DI DIESEL ENGINE

Study of Combustion is one of the complicated phenomenon which involves a chemical reaction at certain elements of fuel like Hydrogen and Carbon combine with Oxygen liberating heat energy and causing and increase in temperature of the gases, within the engine cylinder. IC engines involve complex fluid dynamic interactions between air flow, fuel injection, moving geometries, and combustion. Fluid dynamics phenomena like jet formation, wall impingement with swirl and tumble, and turbulence production are critical for high efficiency engine performance and meeting emissions criteria. The design problems that are encountered include port-flow design, combustion chamber shape design, variable valve timing, injection and ignition timing, and design for low or idle speeds. In Internal combustion Engines CFD may be useful to study one of the most complicated process of combustion in Engines. In this paper a CFD in-cylinder combustion analysis are carried out on a single cylinder four stroke Direct Injection (DI) Engine. The engine is 661cc Kirloskar engine. One of the main process in CFD Analysis is the Grid Generation and defining Time step for solving the problem. Proper and improve grids will give better and accurate results, Grid Independence study has been carried out. The optimum grid is find out in previous work current work is to find optimum time step for CFD analysis three cases are studied with variation in time step from TS 1, TS 0.5 and TS 0.25 the computed data is validated with Experimental data available which are in good agreement.

An anisotropic, fully adaptive algorithm for the solution of convection-dominated equations with semi-Lagrangian schemes

We present in this paper an anisotropic, fully adaptive spatial–temporal algorithm for the solution of convection-dominated equations in the context of semi-Lagrangian schemes. We devise the algorithm within a finite element framework suitable for higher-order finite elements, and derive a newly proposed a posteriori error indicator which allows us to control the local or truncation error in the L2-norm at each time step. This a posteriori error is split into temporal and spatial contributions, leading us to define an optimal time step size and an optimal triangulation, respectively. As regards the spatial adaptation, anisotropic, unstructured triangular meshes are used to capture the distinctive features of the evolving discrete solution of the governing equations. For solutions exhibiting strong anisotropies, the orientation, shape and size of the mesh triangles are provided by a metric tensor valid for linear and quadratic finite elements.Finally, we show the capabilities of the algorithm, for linear and quadratic finite elements, by a series of two- and three-dimensional benchmarks taken from the literature, involving purely convective as well as convection-dominated problems.

Numerical Investigations of Spray Droplet Parameters in a Direct Injection Diesel Engine Using 3-Z Extended Coherent Flame Model

Diesel engine combustion modeling presents a challenging task as the injection starts with the spray formation and breakup of spray into droplets. The computation involved in predicting the in-cylinder fluid mixture during combustion using eulerian and lagrangian approach is rather a cumbersome task. In this work, 3D-CFD computations are performed to understand the behaviour of spray droplet variables on combustion process and emissions in a direct injection diesel engine. The study involves the computation of turbulent flow-field quantities, modelling various processes such as fuel spray distribution, atomization, collision, evaporation, combustion and pollutant formation using a commercial CFD code. Grid independence and time independent studies are carried out for finding the optimum grid size and time step. The numerical results predicted using CFD code is validated with the experimental data available from the literature. The process of combustion and emission characteristics is investigated numerically with respect to spray characteristics. The work is further extended to study the effect of swirl ratio and injection timing on droplet parameters.

On-line reactivity calculation using Lagrange method

In this paper, a novel multi-step method is proposed for solving the inverse point kinetics problem using Lagrange polynomial method. By use of this approach, the need for nuclear power history or the Laplace transform is vanished. Furthermore, the accuracy of the method is of order hn for the (n+1)-point formula, where h is the computational time-step.The three- and five-point formulas of the Lagrange method are used for on-line reactivity calculations and results are benchmarked against reference solutions for different nuclear power forms. Moreover, results for different computational time-steps are compared in each case.The results show the accuracy of the proposed method in all benchmarking cases. For slow transients (large reactor periods), it is shown that time-steps of up to 1s lead to highly reliable reactivity calculations. However, the optimal time-step in almost all cases is shown to be 0.1s. The main advantage of the proposed approach, in contrast with previous numerical methods, is its stability and convergence in large time-step calculations.The proposed method can be used as real time reactivity meter in all nuclear reactors without limitation of nuclear power form.

Compatibility analysis of macroscopic and microscopic traffic simulation modeling

To implement model predictive traffic control to reduce congestion, traffic state variables such as flow, speed, and density need to be accurately predicted with real-time measurements. To evaluate the accuracy of online prediction of a macroscopic traffic model, this paper compares the predicted flow, density, and speed from a macroscopic simulation model with those from a microscopic simulation model, using METANET and VISSIM respectively, on a section of urban freeway. Three levels of traffic demands and seven different time step lengths in macroscopic simulation were applied to evaluate the compatibility of the two models. It was concluded that in the macroscopic simulation model there exists an optimum time step length, under moderate to heavy traffic demands the predicted traffic states from the macroscopic simulation are consistent with the outputs from the microscopic simulation, and under stop-and-go traffic states significant difference exists between the two models.

Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Inverse mask synthesis is achieved by minimizing a cost function on the difference between the output and desired patterns. Such a minimization problem can be solved by a level-set method where the boundary of the pattern is iteratively evolved. However, this evolution is time-consuming in practice and usually converges to a local minimum. The velocity of the boundary evolution and the size of the evolution step, also known as the descent direction and the step size in optimization theory, have a dramatic influence on the convergence properties. This paper focuses on developing a more efficient algorithm with faster convergence and improved performance such as smaller pattern error, lower mean edge placement error, wider defocus band, and higher normalized image log slope. These improvements are accomplished by employing the conjugate gradient of the cost function as the evolution velocity, and by introducing an optimal time step for each iteration of the boundary evolution. The latter is obtained from an extended Euler time range by using a line search method. The authors present simulations demonstrating the efficacy of these two improvements.