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Simulation
Published in Devendra K. Chaturvedi, ®, 2017
The second- and fourth-order Runge–Kutta method is more accurate than the Euler’s method; they still can become unstable, if the step size h is too large. For example, for the one-dimensional linear system, the fourth-order Runge–Kutta method gives a solution which diverges from exact solution for step sizes in the range h > 2.785/c as opposed to h > 2/c for Euler’s method. The computational efficiency of fourth-order Runge–Kutta method can be significantly improved by allowing the step size to vary as the solution progresses. For example, for certain values of x, the function f(t, x) may change very rapidly, in which case small steps are needed. However, in the other regions the function f(t, x) might be quite smooth, even flat, in which case much larger steps are permitted. The key to implementing adaptive step size control is to develop an estimate of local truncation error. The step size can then be decreased or increased dynamically during execution in order to maintain a desired level for error. There are many methods presented by Schilling and Harris (2000) for making adaptive step size algorithms such as interval halving, Runge–Kutta–Fehlberg, and step size adjustment using local error criterion.
Lumped Capacity Transient Heat Transfer
Published in Randall F. Barron, Gregory F. Nellis, Cryogenic Heat Transfer, 2017
Randall F. Barron, Gregory F. Nellis
Most software include native ODE solvers that utilize techniques that are more sophisticated than those discussed in the preceding sections. Most of these native ODE solvers also utilize adaptively changing time steps. The Euler and RK4 techniques discussed in Sections 3.4.1 and 3.4.2 were implemented using a fixed duration time step. This method is not efficient because there are regions of time during the simulation where the solution is not changing substantially, and therefore, large time steps could be taken with little loss of accuracy. Adaptive step-size solutions adjust the size of the time step used based on the local truncation error.
Computational Numerical Methods
Published in Timothy Bower, ®, 2023
A problem is stiff if the solution contains multiple terms with different step size requirements. A slower moving component may dominate the solution; however, a less significant component of the solution may change much faster. Adaptive step size algorithms must take small steps to maintain accurate results. Because stiff ODEs problems have solutions with multiple terms, they come from either second or higher order ODEs or from systems of ODEs.
A beetle antennae search algorithm based on Lévy flights and adaptive strategy
Published in Systems Science & Control Engineering, 2020
In view of the shortcomings of BAS algorithm when solving complex optimization problems, such as low convergence precision, easy to fall into local optimum, and excessive dependence on parameter settings. This paper proposes an algorithm called beetle antennae search algorithm based on Lévy flights and adaptive strategy (LABAS). Firstly, the population used by the algorithm and the corresponding strategy of updating the population using elite individual information enhance its optimization ability, stability and exploitation ability. Secondly, the Lévy flights and the scaling factor improve the ability of the algorithm to explore the region of the global optimal solution, avoiding falling into local optimum and converge to the global optimal value more quickly. Thirdly, the adaptive step size strategy avoids the difficulty of parameter setting and can be automatically adjusted according to the type and size of the problem. Fourthly, GOBL enhances the diversity of the population and also makes the algorithm has better ability to find the optimal solution. The above improvements balance the global exploration and local exploitation of the algorithm to some extent. Simulation experiments and comparative analysis show that the LABAS algorithm is superior to the BAS algorithm and other comparison algorithms in terms of accuracy, convergence rate, stability, robustness and local optimal value avoidance. In our future research work, the LABAS algorithm will be applied in the optimization problems in the textiles, carbon fibre production and other fields.
Modeling and simulation of single droplet drying in an acoustic levitator
Published in Drying Technology, 2023
Martin Doß, Nadja Ray, Eberhard Bänsch
To discretize our model in time, we use the backward Euler (BDF1) method. The adaptive step size is determined automatically by the simulation software COMSOL Multiphysics® with respect to a given maximum. More precisely, we require s for all time steps. The implicit time-stepping problems are then solved as fully coupled systems using Newton’s method with an adaptive damping factor. In each Newton step, the direct solver PARDISO is applied to the linearized system.
Adaptive step-size forward advection method for aerosol process simulation
Published in International Journal of Digital Earth, 2023
Yuang Wu, Shuo Liu, Bowen Shu, Weichao Sun, Sheng Wang, Hongyang Zhang, Chenchen Chen
In the comparison experiment between the forward adaptive step size and the fixed step size (Figures 23 and 24, and Table 3), we found that the overall accuracy (Kappa) of the adaptive step size method was better than that of the fixed time step.