Modified Euler Method Research Articles

Cube Arithmetic: Improving Euler Method for Ordinary Differential Equation using Cube Mean

The Euler method is a first-order numerical procedure for solving Ordinary Differential Equation (ODEs) problems. It is an effective and easy method to solve initial value problems. Although Euler provides simple procedure for solving ODEs, there have been issues such as complexity, time of processing and accuracy that compelled the use of other, more complex, methods. Improvements to the Euler method have attracted much attention resulting in numerous modified Euler methods. This paper proposes Cube Arithmetic, a modified Euler method with improved accuracy. The efficiency of Cube Arithmetic was compared with Euler Arithmetic and tested using SCILAB against exact solutions. Results indicate that not only Cube Arithmetic provided solutions that are similar to exact solutions at small step size, but also at higher step size, hence producing more accurate results.

Wake and structure model for simulation of cross-flow/in-line vortex induced vibration of marine risers

Three dimensional responses of riser subjected to Vortex Induced Vibration (VIV) are investigated. Proportionality relations of stress and fatigue damage are mentioned. A computer code has been developed for time domain modeling of VIV of riser accounting for both Cross-Flow (CF) and In-Line (IL) vibration. The wake oscillator model is used to calculate the VIV of each strip. The wake oscillators are coupled to the dynamics of the long riser, while the Newmark-beta method is used for evaluating the structural dynamics of riser. The wake dynamics, including IL and CF vibrations, is represented using a pair of non-linear Van der Pol equations that solved using modified Euler method. The existing experimental and numerical results for stepped and sheared current are used to validate the proposed model and the results show reasonable agreement. The proposed model was implemented on Amir-Kabir semi-submersible riser deployed at the water depth of 713 meters of Caspian Sea. CF/IL VIV of this riser is simulated for various current velocities. The results show that although displacement amplitude of IL direction is lower than CF direction but because of higher curvature, stress values of IL direction for some cases can be higher than CF direction. Also because of higher frequency of IL direction, fatigue damage of this direction can be higher than CF one in some cases. It is shown that with increasing of current velocity; however, variation of displacement amplitude of two directions is low but stress increased and fatigue damage also increased with higher rate. For lower velocities which the modes are controlled with tension, stress and fatigue damage of IL direction is higher than CF direction.

Constrained LMMSE‐based object‐specific reconstruction in compressive sensing

In many applications like surveillance, it is essential to detect the presence of specific objects by seeing an image or video without being concerned about details of the scene. The reconstruction algorithms proposed in the compressive sensing literature try to iteratively reconstruct the full image. Hence, those are computationally expensive. If some prior knowledge of the object is available, then a closed-form reconstruction algorithm can be formulated. The goal is to reconstruct the object of interest efficiently without being bothered about the quality of reconstruction of the scene. To address this situation, a constraint is formulated and incorporated into linear minimum mean square error (LMMSE) estimator to form a closed-form solution. This compact solution is capable of meaningfully reconstructing the object relative to the scene. In the proposed method, an ill-conditioned matrix inversion problem has been faced and overcome by regularization method. To boost the speed of the algorithm, a modified Euler method is proposed for finding the regularization parameter. To further speedup the reconstruction process, larger images are divided into several pieces and each piece is reconstructed separately using constrained LMMSE. For a given number of measurements, the simulation-based results demonstrate acceptable quality of reconstruction with minimal computational effort.

Large Multi-Machine Power System Simulations Using Multi-Stage Adomian Decomposition

Multi-stage Adomian decomposition method (MADM) is a proven semi-analytical approximation solution technique for ordinary differential equations (ODEs), which provides a rapidly convergent series by integrating over multiple time intervals. Applicability of MADM for large nonlinear differential algebraic systems (DAEs) is established for the first time in this paper using the partitioned solution approach. Detailed models of power system components are approximated using MADM models. MADM applicability is verified on 7 widely used test systems ranging from 10 generators, 39 buses to 4092 generators, 13659 buses. Impact of the step size and the number of terms is investigated on the stability and accuracy of the method. An average speed up of 42% and 26% is observed in the solution time of ODEs alone using the MADM when compared to the midpoint-trapezoidal (TrapZ) method and the modified-Euler (ME) method, respectively. MADM accuracy is found to be similar to the ME and comparable to the TrapZ method. MADM stability properties are found to be better than the ME and weaker than the TrapZ method. An average speed up of 13% and 5.85% is observed in the overall solution time using MADM w.r.t. TrapZ and ME methods, respectively.

Perturbation nonlinear response of tension leg platform under regular wave excitation

Conceptual discussion on highly nonlinear Duffing type equation of surge motion of TLP gives a deep view on structural response under environmental loads with some simplifications. Such analytical response is a simple form that clarifies important points in behavior of the structure. This paper presents the dynamic motion responses of a TLP in regular sea waves obtained by applying three methods in time domain using MATLAB software. Surge motion equation of TLP is highly nonlinear because of large displacement and it should be solved with large perturbation parameter in time domain. In this paper, homotopy perturbation method (HPM) is used to solve highly nonlinear differential equation of surge motion. Also the numerical methods such as modified Euler method (MEM) and MATLAB are used to solve the ordinary differential equation of motion. MEM is used for highly nonlinear Duffing type equation of surge motion of TLP, whereas the nonlinear part is not considered in MATLAB solution. Calculated responses by analytical HPM are compared with those obtained from both MEM and MATLAB responses.

Modified Euler approximation of stochastic differential equation driven by Brownian motion and fractional Brownian motion

ABSTRACTConsider a class of mixed stochastic differential equation (SDE) involving both a Brownian motion and a fractional Brownian motion with Hurst parameter We get the mean square rate of convergence by using a modified Euler method, here δ is the diameter of partition. As we know, the classical Euler method has the rate of convergence for mixed SDE and δ2H − 1 (resp. δH) for pathwise (resp. Skorokhod) SDE driven only by fBm, which were proved by Mishura and Shevchenko Mishura and Shevchenko (2011) and Mishura and Shevchenko (2008), respectively. Therefore, we obtain a better result than those of them.

Numerical and experimental study on the dynamic behavior of a sea-star tension leg platform against regular waves

This paper describes an experimental work on a 1: 100 scaled model of a miniature sea-star tension leg platform (TLP) in a wave flume. Two different numerical models are developed: finite element model (FEM) based on the Morison equation and boundary element model (BEM) based on a 3D diffraction/radiation theory. The developed codes are used to calculate hydrodynamic forces and related coefficients. The nonlinear hull/tendon coupled dynamic equation of a mini seastar TLP is solved by using a modified Euler method (MEM). The results of numerical modeling of the motion response behavior of the platform in different degrees of freedom are compared with experimental data. This comparison shows good agreement between the results. Furthermore, this modeling reveals that the first-order diffraction method and quasi-static tendon modeling are sufficient in general for the hydrodynamic analysis of the sea-star TLP.

Numerical Solution of Impulsive Fuzzy Initial Value Problem by Modified Euler's Method

In this paper we present numerical solution of first order impulsive fuzzy differential equation by modified Euler's method for the first time. The algorithm is discussed in detail and the result of the numerical solution is given by solving the numerical example and the graph is plotted finally.

Evolution of density and velocity profiles of matter in large voids

We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark energy - are continuous media with ideal fluid energy-momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations in the comoving to cosmological background reference frame for every component are obtained from equations of conservation and Einstein's ones and are integrated by modified Euler method. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is mush larger than the particle horizon. Results show how the profiles of density and velocity of matter in spherical voids with different overdensity shells are formed.

A Numerical Method for Solving Matrix Coefficient Heat Equations with Interfaces

AbstractIn this paper, we propose a numerical method for solving the heat equations with interfaces. This method uses the non-traditional finite element method together with finite difference method to get solutions with second-order accuracy. It is capable of dealing with matrix coefficient involving time, and the interfaces under consideration are sharp-edged interfaces instead of smooth interfaces. Modified Euler Method is employed to ensure the accuracy in time. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner. Extensive numerical experiments illustrate the feasibility of the method.

Lattice Boltzmann analysis of micro-particles transport in pulsating obstructed channel flow

Dispersion and deposition of microparticles are investigated numerically in a channel in the presence of a square obstacle and inlet flow pulsation. Lattice Boltzmann method (LBM) is used to simulate the flow field and modified Euler method is employed to calculate particles trajectories with the assumption of one-way coupling. The forces of drag, gravity, Saffman lift and Brownian motion are included in the particles equation of motion. The effects of pulsation amplitude (AMP), Strouhal number and particles Stokes number (Stk) are rigorously studied on particles dispersion and deposition efficiency. Flow vortex shedding and particles dispersion patterns together with the averaged fluid–particle relative velocity and deposition efficiency plots are all discussed thoroughly. The results show that increment of pulsation amplitude enforces the vortices to form closer to the obstacle until their shape deteriorates as Strouhal number ratio (SNR) rises. The average recirculation length shrinks to its minimum at each studied Amp when SNR escalates to 2. Various behaviors are categorized for dispersion pattern of particles when Stokes number changes from 0.001 to 4. Deposition efficiency is indirectly related to Amp for Stk≤2 while for higher Stokes numbers (2<Stk≤4) they show direct relationship. Deposition pattern becomes rather independent of SNR at Amp=0.1. The grid independency test was performed for the LBM analysis, and simulation code was successfully verified against credible benchmarks.

Versatile Muskingum flood model with four variable parameters

Early researchers recognised that the long-established Muskingum hydrologic routing model's parameters vary with flood characteristics, but no methods were developed to account for this variation. Such methods would have made the model too complex to solve using prevailing technical knowledge. This paper presents a versatile non-linear Muskingum model with four variable parameters; each is represented by a two-step function of a dimensionless inflow variable, so the model thus has eight parameters. The model is general and produces a wide array of 15 special models with the number of parameters ranging from four to seven. The routing procedure is based on the modified Euler method. Three examples with different hydrograph types were used to evaluate model performance. Based on the application results, guidelines for model selection for different hydrograph types are presented. The recommended models have four to six parameters and substantially improve model performance for predicting flood flows compared with the traditional three-parameter model. This research continues to promote alternative thinking to improve model performance by modifying model structure, instead of the solution algorithm.

A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

SUMMARYThis paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Development of the distinct lattice spring model for large deformation analyses

SUMMARYThis study develops the distinct lattice spring model (DLSM) for geometrically nonlinear large deformation problems. The formulation of a spring bond deformation under a large deformation is derived under the Lagrange framework using polar decomposition. The results reveal that the DLSM's stiffness matrix under small deformations is the tangent stiffness matrix of the DLSM under large deformations. The formulation of the spring bond internal force under a given configuration is also presented and can be used to calculate the unbalanced force. Using these formulations, three nonlinear solving methods (the Euler method, modified Euler method, and Newton method) are developed for the DLSM with which to tackle large deformation problems. To investigate the performance of the developed model, three numerical examples involving large deformations are presented, the results of which are also in good agreement with the analytical and finite element method solutions. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

The Euler–Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler’s method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does—in contrast to classical Monte Carlo methods—not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case.

A predictor–corrector iterated Tikhonov regularization for linear ill-posed inverse problems

Many questions in science and engineering often give rise to linear ill-posed inverse problems. To enable meaningful approximate solutions of the inverse problems, various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations. One of the most popular regularization techniques is iterated Tikhonov regularization, which has attracted considerable attention due to its interesting applications in practical. However, this regularization often suffers from inaccurate results and low computational efficiency in some situations. In this paper, we proposed an accelerated predictor–corrector iterated Tikhonov regularization. This method combined the classical iterated Tikhonov regularization with modified Euler method, which could improve computational efficiency without sacrificing numerical accuracy. The convergence rate of the accelerated regularization was investigated theoretically when the right-hand side was corrupted by noise. In particular, we derived an error estimate of optimal order for the convergence of the accelerated regularization. Both one-and two-dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated Tikhonov regularization.

Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step

AbstractCalculation of actual crop evapotranspiration under soil water stress conditions is crucial for hydrological modeling and irrigation water management. Results of actual evapotranspiration depend on the estimation of water stress coefficient from soil water storage in the root zone, which varies with numerical methods and time step used. During soil water depletion periods without irrigation or precipitation, the actual crop evapotranspiration can be calculated by an analytical method and various numerical methods. We compared the results from several commonly used numerical methods, including the explicit, implicit and modified Euler methods, the midpoint method, and the Heun's third‐order method, with results of the analytical method as the bench mark. Results indicate that relative errors of actual crop evapotranspiration calculated with numerical methods in one time step are independent of the initial soil water storage in the range of soil water stress. Absolute values of relative error decrease with the order of numerical methods. They also decrease with the number of time step, which can ensure the numerical stability of successive simulation of soil water balance. Considering the calculation complexity and calculation errors caused by numerical approximation for different time step and maximum crop evapotranspiration, the explicit Euler method is recommended for the time step of 1 day (d) or 2 d for maximum crop evapotranspiration less than 5 mm/d, the midpoint method or the modified Euler method for the time step of up to one week or 10 d for maximum crop evapotranspiration less than 5 mm/d, and the Heun's third‐order method for the time step of up to 15 d. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Effects of variable fluid properties on unsteady thin-film flow over a non-linear stretching sheet

The effects of the variation of viscosity and surface tension with temperature on unsteady flow of a thin viscous liquid film over a heated horizontal stretching surface are analyzed considering general form of the stretching velocity and the temperature distribution. Using perturbation technique, an evolution equation is derived from the governing equations and this evolution equation (non-linear PDE) is solved numerically by using a combination of modified Euler method, the Newton–Kantorovich method, and the finite difference method. This leads to a five-parameter problem for some representative value of the parameters. It is shown that the film thickness decreases with the decrease of viscosity of the fluid as temperature increases and more or less heat flows out of the liquid through the stretching surface. Film thins faster both for thermo-capillary force and decrease of viscous force provided temperature decreases along the stretching direction. Physical explanations are furnished where it is necessary to justify the results.

Modeling and simulation of a class of liquid propellant engine pressurization systems

In this paper, a new approach for modeling a class of pressurization systems for liquid propellant engines is presented. High pressure gas stored in special capsules is injected in a controlled way onto the gas volume on top of the propellant tanks. Instantaneous pressure and temperature values of propellant tanks and the capsules are computed. The computed constitutive values are compared with experimental results. The modeling approach accounts for variations of pressure and density of the pressurant inside the capsules and propellant tanks. It also accounts for heat transfer between pressurant and its surroundings. Equations for determination of the pressurant shut-down time are developed. Similarly, equations for determination of the propellant pump inlet pressure are also derived. The final governing equations are in terms of a set of differential equations that are numerically solved using the modified Euler method. The primary application of this research is to determine the pressurization system exit pressure. In case of turbo-pump feed systems, this work may be used to detect the occurrence of the cavitation phenomenon in the pumps. The method is limited to: (1) thermally insulated propellant tanks and (2) use of several storage capsules that are made of insulating composite material (i.e., exit mass flow rate is small, pressurant is stored at high pressure and low volume, temperature variations are small, and pressure drop in the capsules is not appreciable).

Numerical Stability of Existing Monte Carlo Burnup Codes in Cycle Calculations of Critical Reactors

We show that major existing Monte Carlo burnup codes are numerically unstable in cycle calculations of critical reactors; spatial oscillations of the neutron flux can be observed even when relatively small time steps are used. This is caused by using the explicit Euler or midpoint method that appear to be numerically unstable with the step sizes common in cycle calculations. More stable methods that are common in deterministic burnup calculations, like the modified Euler method, can easily be introduced into the Monte Carlo burnup codes.