First-order Euler Scheme Research Articles

Hybrid predictor-corrector numerical schemes: λ-tracking, sampled-data observers and data assimilation

This paper considers hybrid ordinary differential equation (ODE) solvers for process dynamics constructed by combining standard numerical schemes with standard observers. Specifically, we combine the first-order Euler scheme with a Luenberger observer. The key ideas are to take advantage of available process output information and to switch from the numerical scheme to the process output-driven observer when the numerical scheme alone would produce inadequate results. Within this setup, two tasks emerge: How to choose the observer gain? How to choose the step size in the numerical scheme? Underpinning our approach is a λ tracking-based sampled-data observer that invokes a λ dead zone. This λ tracking observer determines the observer gain and the numerical step-size adaptively. The resulting adaptive hybrid algorithm is a time-stepping numerical scheme. Using a sampled-data observer allows for process measurements to be only available at some discrete times, whilst adaptive tuning allows the gains and sampling times to adjust automatically to each other – rather than both being subjected to designer's choice. Results are illustrated with examples of simulation.

Feedback Integrators for Mechanical Systems with Holonomic Constraints.

The feedback integrators method is improved, via the celebrated Dirac formula, to integrate the equations of motion for mechanical systems with holonomic constraints so as to produce numerical trajectories that remain in the constraint set and preserve the values of quantities, such as energy, that are theoretically known to be conserved. A feedback integrator is concretely implemented in conjunction with the first-order Euler scheme on the spherical pendulum system and its excellent performance is demonstrated in comparison with the RATTLE method, the Lie–Trotter splitting method, and the Strang splitting method.

On the steady laminar boundary-layer flow over the family of rotating tori

We present the laminar boundary layer flow over the standard tori rotating in an otherwise quiescent incompressible fluid. The mean flow equations are derived by using a specific form of the toroidal–poloidal coordinate system. A pseudo-spectral collocation method mixed with the first-order Euler scheme is employed to solve the related steady mean flow equations. A fixed aspect ratio R is defined as the ratio of radii of the torus and its tube that distinguishes each fixed torus within the general family of standard tori. Tori of spindle type turn into a sphere when the aspect ratio R=0 and thus the mean flow results for the rotating sphere are consistently reproduced. The influence of aspect ratio R on the steady mean velocity components and the wall shear stresses are presented in detail. In a precise manner, the mean flow results for each rotating torus are explainable in a consistent way to the already established case of steady laminar boundary layer flow over the rotating sphere. The significance of the current work is discussed in terms of the hydrodynamic instabilities of boundary layer flows over the family of rotating tori.

First-order Euler scheme for SDEs driven by fractional Brownian motions: The rough case

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac{1}{3} \frac{1}{2}$. The current contribution generalizes the modified Euler scheme to the rough case $\frac{1}{3}<H<\frac{1}{2}$. Namely, we show a convergence rate of order $n^{\frac{1}{2}-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations

Explicit integrators for real-time propagation of time-dependent Kohn-Sham equations are compared regarding their suitability for performing large-scale simulations. Four algorithms are implemented and assessed for both stability and accuracy within a plane-wave pseudopotential framework, employing the adiabatic approximation to the exchange-correlation functional. Simulation results for a single sodium atom and a sodium atom embedded in bulk magnesium oxide are discussed. While the first-order Euler scheme and the second-order finite-difference scheme are unstable, the fourth-order Runge-Kutta scheme is found to be conditionally stable and accurate within this framework. Excellent parallel scalability of the algorithm up to more than a thousand processors is demonstrated for a system containing hundreds of electrons, evidencing the suitability for large-scale simulations based on real-time propagation of time-dependent Kohn-Sham equations.

Transient Schemes for Capturing Interfaces of Free-Surface Flows

This article presents a new methodology for the development of Transient Interpolation for Capturing of Surfaces schemes suitable for the simulation of free-surface flows, which is given the acronym TICS. The newly developed approach is based on a switching strategy that combines a bounded high-order transient scheme with a bounded compressive transient scheme. Bounded high-order and compressive transient schemes are constructed by discretizing the transient term in the volume-of-fluid (r) equation over a temporal control-volume in a way similar to the discretization of the convection term over a spatial control-volume, allowing advances in building convective schemes to be exploited in the development of bounded high-order and compressive transient schemes. Following that approach, a bounded version of the second-order upwind Euler scheme is constructed (B-SOUE). The B-SOUE is used to develop a family of temporal compressive schemes that is denoted by the B-CEm family, where “m” refers to the slope of the scheme on a temporal normalized variable diagram. The TICS methodology is then applied to the B-SOUE scheme and the B-CEm family of schemes to create a new family of transient interface-capturing schemes that is designated by TICSm. The virtues of the TICSm family, in producing a steep interface for the volume-of-fluid (r) field that defines the volume fraction occupied by the different fluids in a computational domain, are demonstrated through results generated using two schemes of the family (TICS1.75 and TICS2.5). The accuracy of the new transient TICS schemes is compared to the first-order Euler scheme, the Crank-Nicolson scheme, and the B-SOUE scheme by solving four pure advection test problems (advection of hollow shapes in an oblique flow field and advection of a solid body in a rotational flow field) and one flow problem (the break of a dam) using both the SMART and the STACS convective schemes. Results, displayed in the form of interface contours, demonstrate that predictions obtained with TICS1.75 and TICS2.5 are far more accurate and less diffusive, preserving interface sharpness and boundedness at all Courant number values considered.

Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution

Abstract The simulation of transient flows is relevant in several applications involving viscoelastic fluids. In the last decades, much effort has been spent on deriving time-marching schemes able to efficiently solve the governing equations at low computational cost. In this direction, decoupling schemes, where the global system is split into smaller subsystems, have been particularly successful. However, most of these techniques only work if inertia and/or a large Newtonian solvent contribution is included in the modeling. This is not the case for polymer melts or concentrated polymer solutions. In this work, we propose two second-order time-integration schemes for discretizing the momentum balance as well as the constitutive equation, based on a Gear and a Crank–Nicolson scheme. The solution of the momentum and continuity equations is decoupled from the constitutive one. The stress tensor term in the momentum balance is replaced by its space-continuous but time-discretized form of the constitutive equation through an Euler scheme implicit in the velocity. This adds velocity unknowns in the momentum equation thus an updating of the velocity field is possible even if inertia and solvent viscosity are not included in the model. To further reduce computational costs, the non-linear relaxation term in the constitutive equation is taken explicitly leading to a linear system of equations for each stress component. Four benchmark problems are considered to test the numerical schemes. The results show that a Crank–Nicolson based discretization for the momentum equation produces oscillations when combined with a Crank–Nicolson based scheme for the constitutive equation whereas, if a Gear based scheme is implemented for the constitutive equation, the stability is found to be dependent on the specific problem. However, the Gear based scheme applied to the momentum balance combined with both second-order methods used for the constitutive equation is stable and accurate and performs much better than a first-order Euler scheme. Finally, a numerical proof of the second-order convergence is also carried out.

Improving the efficiency of BD algorithms for biological systems simulations

In this work, an error analysis methodology is applied to the study of various integration schemes used in Brownian Dynamics simulations of ion channels. Three algorithms have been compared for the integration of the full Langevin Equation [1]. A first-order Euler scheme [2], the Verlet-like algorithm proposed by [3], and a novel Predictor/Corrector (PC) scheme [4] have been implemented and analyzed using our assessment methodology [5]. Our results show that a significant increase in the integration timestep, and a subsequent reduction in computational cost and improvement in efficiency, can be achieved with the second order Verlet-like and PC schemes, while maintaining an excellent accuracy for the description of both structure and dynamics of the electrolyte solution. This work extends the analysis developed in [5] to dynamic properties of the solution as well as structural aspects of inhomogeneous systems such as a lipid membrane.

AN AUTOMATIC LOAD STEPPING ALGORITHM WITH ERROR CONTROL

This paper presents an algorithm for controlling the error in non-linear finite element analysis which is caused by the use of finite load steps. In contrast to most recent schemes, the proposed technique is non-iterative and treats the governing load–deflection relations as a system of ordinary differential equations. This permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error. The latter is measured by computing the difference between two estimates of the displacement increments for each load step, with the initial estimate being found from the first-order Euler scheme and the improved estimate being found from the second-order modified Euler scheme. If the local truncation error exceeds a specified tolerance, then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation. Local extrapolation is also used to predict the size of the next load step following a successful update. In order to control not only the local load path error, but also the global load path error, the proposed scheme incorporates a correction for the unbalanced forces. Overall, the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step. Because the solution scheme is non-iterative and founded on successful techniques for integrating systems of ordinary differential equations, it is particularly robust. To illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance, detailed results are presented for a variety of elastoplastic boundary value problems.

Schemes and Parameters in 2D Vortex Simulation

Dependence of flow features in two-dimensional (2D) vortex simulation on time-integration scheme is studied. Three types of scheme are applied to a flow past an impulsively started flat plate set normal to the freestream; the first-order Euler scheme, the second-order Adams-Bashforth scheme and the fourth-order Runge-Kutta scheme. Calculated results are compared with experiments. The Adams-Bashforth scheme and the Runge-Kutta scheme prove to give flow features close to each other, while the Euler scheme gives somewhat different results from the others. As to the temporal growth of separation bubbles, the Euler scheme gives the best agreement with experiments. Effects of parameters such as the numerical core-radius and the time-step are also studied.