Explicit Euler Method Research Articles

Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.

Explicit time integration of transient eddy current problems

SummaryFor time integration of transient eddy current problems, commonly implicit time integration methods are used, where in every time, step 1 or several nonlinear systems of equations have to be linearized with the Newton‐Raphson method because of ferromagnetic materials involved. In this paper, a generalized Schur complement is applied to the magnetic vector potential formulation, which converts a differential‐algebraic equation system of index 1 into a system of ordinary differential equations with reduced stiffness. For the time integration of this ordinary differential equations system of equations, the explicit Euler method is applied. The Courant‐Friedrich‐Levy stability criterion of explicit time integration methods may result in small time steps. Applying a pseudoinverse of the discrete curl‐curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudoinverse, the preconditioned conjugate gradient method is used. The cascaded subspace extrapolation method is presented to produce suitable start vectors for these preconditioned conjugate gradient iterations. The resulting scheme is validated using the nonlinear TEAM 10 benchmark problem.

A finite difference method for a conservative Allen–Cahn equation on non-flat surfaces

We present an efficient numerical scheme for the conservative Allen–Cahn (CAC) equation on various surfaces embedded in a narrow band domain in the three-dimensional space. We apply a quasi-Neumann boundary condition on the narrow band domain boundary using the closest point method. This boundary treatment allows us to use the standard Cartesian Laplacian operator instead of the Laplace–Beltrami operator. We apply a hybrid operator splitting method for solving the CAC equation. First, we use an explicit Euler method to solve the diffusion term. Second, we solve the nonlinear term by using a closed-form solution. Third, we apply a space–time-dependent Lagrange multiplier to conserve the total quantity. The overall scheme is explicit in time and does not need iterative steps; therefore, it is fast. A series of numerical experiments demonstrate the accuracy and efficiency of the proposed hybrid scheme.

The Discrete Adjoint Gradient Computation for Optimization Problems in Multibody Dynamics

The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

Enhanced Discrete-Time Modeling via Variational Integrators and Digital Controller Design for Ground Vehicles

In this paper, a discrete-time controller for ground vehicles is proposed, for the tracking of desired references for lateral and yaw velocities. A novel discrete-time model of the vehicle dynamics is considered, which is obtained by means of a variational integrator known as symplectic Euler. The control inputs are given by active front steering and rear torque vectoring. Simulations have been carried out considering a virtual automobile implemented in CarSim, and a comparative study has been performed considering alternative discrete-time controllers, based on a sampled continuous-time controller and on a discrete-time vehicle model obtained by means of the explicit Euler method. The controller performance has been analyzed considering various sampling periods, showing a robustness property when the sampling period increases. Moreover, the robustness of the proposed discrete-time controller performance has been tested to show its robustness versus parameter variations.

Justification of Godunov’s scheme in the multidimensional case

The classical Godunov scheme for the numerical solution of 3D gas dynamics equations is justified in the multidimensional case. An estimate is obtained for the error induced by replacing the exact solution of the multidimensional discontinuity breakup problem (known as the Riemann problem) with the solution of the 1D problems with the data on the left and right of the interface of each cell without considering the complicated flow in the neighborhood of the cells’ vertices. It is shown that, in the case of plane interfaces, the error has the first order of smallness in the time step and the approximate solution converges to the solution of semidiscrete equations as the time step vanishes. In fact, the time integration of these equations using the explicit Euler method represents the Godunov scheme.

Structural deformation behavior of Jacquardtronic lace based on the mass-spring model

To simulate the structural deformation behavior of Jacquardtronic lace, which is formed on a multibar Jacquard Raschel machine and is widely used in female fashion, the key influencing factors on structural deformation were introduced and a tailored mass-spring model was built in which the lace was considered to be numbers of evenly distributed particles connected by elastane springs. These springs covered structural springs, restriction springs and spiral springs, respectively, created for ground pillars, Jacquard inlays, pattern lapping and elastane yarns. The elastane force on particles was analyzed to study motion state and deformation behavior with the explicit Euler method. The deformation simulated models were implemented by a simulator program via Visual C++ and were tested with a lace sample. Particular attention was paid to analyzing the effect of Jacquard structures and yarn tension on deformation and the detection model was introduced to avoid improper deformation caused by excessive yarn tension. The simulation results showed practicability and efficiency when compared with the real sample.

Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions

We analyze the implicit Euler discretization for a class of convex linear-quadratic optimal control problems with control appearing linearly. Constraints are defined by lower and upper bounds for the controls, and the cost functional may depend on a regularization parameter ν. Without any structural assumption on the optimal control we prove convergence of order 1 w.r.t. the mesh size for the discrete optimal values. Under the additional assumption that the optimal control is of bang-bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters. By applying this result to the implicit Euler discretization we improve existing error estimates for discretizations based on the explicit Euler method. Numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang-bang controls.

MODELING AND ANALYSIS OF UNSTEADY FLOW BEHAVIOR IN DEEPWATER CONTROLLED MUD-CAP DRILLING

A new mathematical model was developed in this study to simulate the unsteady flow in controlled mud-cap drilling systems. The model can predict the time-dependent flow inside the drill string and annulus after a circulation break. This model consists of the continuity and momentum equations solved using the explicit Euler method. The model considers both Newtonian and non-Newtonian fluids flowing inside the drill string and annular space. The model predicts the transient flow velocity of mud, the equilibrium time, and the change in the bottom hole pressure (BHP) during the unsteady flow. The model was verified using data from U-tube flow experiments reported in the literature. The result shows that the model is accurate, with a maximum average error of 3.56% for the velocity prediction. Together with the measured data, the computed transient flow behavior can be used to better detect well kick and a loss of circulation after the mud pump is shut down. The model sensitivity analysis show that the water depth, mud density and drill string size are the three major factors affecting the fluctuation of the BHP after a circulation break. These factors should be carefully examined in well design and drilling operations to minimize BHP fluctuation and well kick. This study provides the fundamentals for designing a safe system in controlled mud-cap drilling operati.

Education

Euler's method appears early and often in the undergraduate curriculum, not only as a useful method for approximating solutions to differential equations, but also as a scaffold for introducing students to the notion of consistency and accuracy. Interestingly, most students walk away with the impression that Euler, while simple and of first order, is bulletproof in the sense that as long as one is willing to wait, choosing a very small timestep will lead to a reasonable approximation of the solution. Authors Corless and Jankowski douse that notion using a leaky bucket of water in this issue of the Education section. Specifically, they use the relatively mild ODE describing the flow of water from a hole in a bucket as a model problem to explore the method of modified equations. In “Variations on a Theme of Euler,” the authors use their model problem to reintroduce readers to the residual and, from there, the concept of optimal backward error and the method of modified equations for the explicit (forward) Euler and implicit (backward) Euler methods. Using the method of modified equations, one can recast the iterative equation to show that the numerical scheme more closely approximates a modified equation that is different from the original governing equation on which the model is based. In other words, one can think of errors arising from numerical integration as errors that emerge in the modeling process. This article is a perfect module for an undergraduate modeling or numerical analysis course. Since a solid course in either area usually involves both subjects, this article is well aimed and suitable for an audience with broad interests.

Analysis of estimation accuracy of the first moments of a Monte Carlo solution to an SDE with Wiener and Poisson components

In this paper, the estimation accuracy of the first moments of a numerical solution to an SDE with Wiener and Poisson components is investigated by a generalized explicit Euler method. Exact expressions for the mathematical expectation and variance of a test SDE solution are obtained. These expressions allow us to investigate the estimation accuracy obtained by a Monte Carlo method versus the SDE parameters, the integration step, and the size of the ensemble of simulated trajectories of the solution. The results of test numerical experiments are presented.

Simulation of two-dimensional sloshing phenomenon by generalized finite difference method

In this paper, a meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to efficiently and accurately simulate the sloshing phenomenon in a two-dimensional numerical wave tank. When a numerical wave tank is excited horizontally or vertically, the disturbance on the free surface and the flow field in the tank is called sloshing. Based on the theorem of ideal fluid, the mathematical description of the sloshing problem is a time-dependent boundary value problem, governed by a second-order partial differential equation and two non-linear free-surface boundary conditions. In this paper, the GFDM and the explicit Euler method are adopted, respectively, for spatial and temporal discretizations of this moving-boundary problem. After the discretization by the explicit Euler method, the elevation of free surface is updated and a boundary value problem is yielded at every time step. Since the GFDM, a newly-developed domain-type meshless method, can truly get rid of time-consuming meshing generation and numerical quadrature, we adopted the GFDM to efficiently analyze this boundary value problem at every time step. To use the moving-least squares method of the GFDM can express the derivatives as linear combinations of nearby function values, such that the numerical procedures of the GFDM are very simple and efficient. We provided four numerical examples to verify the simplicity and the accuracy of the proposed meshless scheme. In addition, some factors of the proposed numerical scheme are systematically investigated via a series of numerical experiments.

Discrete-Time Modeling and Control of Induction Motors by Means of Variational Integrators and Sliding Modes—Part I: Mathematical Modeling

This paper deals with the discrete-time modeling of induction motors (IMs) by means of a variational integrator. A Lagrangian is first formulated for the IM, where the corresponding discrete Lagrangian is derived based on the one-point quadrature rule integration method (in this case, the variational integrator is known as symplectic Euler). Then, a discrete action principle is applied, which guarantees the optimum path in a discrete-time setting. The forcing and dissipation are added by using the discrete Lagrange d'Alembert principle. Finally, discrete-time update rules are obtained for IMs. Then, theoretic properties such as relative degree and steady state are studied for the continuous-time symplectic Euler and explicit Euler models. Simulations are carried out for different sampling periods, where results are compared with a discrete-time model obtained with the classical explicit Euler method. It is put in evidence that the symplectic Euler model approximates better the IM dynamics than the model obtained with the classical Euler method.

Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems

In this paper, we consider the numerical approximation of a reaction–diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion d and of the reaction timescales given by the real and imaginary parts α and β of the eigenvalues of J(Pe), the Jacobian of the reaction part at the equilibrium point Pe. Focusing on the case α=0,β≠0, we obtain stability regions in the plane (ξ,ν), where ξ=λ(h;d)ht, ν=βht, ht time stepsize, λ lumped diffusion scale depending also from the space stepsize h and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order p=2,4,6. In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge–Kutta method that are symplectic in the absence of diffusion. Hence, by estimating λ, for each method we derive stepsize restrictions ht⪅Fmet(h;d,β,p) in terms of the stability curve Fmet depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka–Volterra PDE system with solutions oscillating only in time for the presence of a centre-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the Turing–Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.

A numerical model of transient thermal transport phenomena in a high-temperature solid–gas reacting system for CO2 capture applications

A numerical model coupling transient radiative, convective, and conductive heat transfer, mass transfer, and chemical kinetics of a heterogeneous solid–gas reacting system has been developed and applied to a model reaction: the decomposition of calcium carbonate into calcium oxide and carbon dioxide. The model reaction is one of two reactions involved in calcium oxide looping, a proposed thermochemical process suitable for use with concentrated solar radiation for the capture of carbon dioxide. The analyzed system is a single, porous particle in an idealized, reactor-like environment that is subjected to concentrated solar irradiation. The finite volume and explicit Euler methods are used to solve volume-averaged governing equations numerically. The model predicts the time-dependent temperature distributions as well as local solid and fluid phase composition. For the baseline simulation, complete decomposition of a 2.5mm radius particle exposed to 1MWm−2 solar irradiation is reached in 35s. The model is further used to investigate physical parameters and operating conditions under which solar-driven calcium oxide looping may be employed for carbon capture. Time to complete conversion decreases under conditions favorable for increased rate of heating, such as optimum particle size and increased incident irradiation.

Implicit solution of the material transport in Stokes flow simulation: Toward thermal convection simulation surrounded by free surface

We present implicit time integration schemes suitable for modeling free surface Stokes flow dynamics with marker in cell (MIC) based spatial discretization. Our target is for example thermal convection surrounded by deformable surface boundaries to simulate the long term planetary formation process. The numerical system becomes stiff when the dynamical balancing time scale for the increasing/decreasing load by surface deformation is very short compared with the time scale associated with thermal convection. Any explicit time integration scheme will require very small time steps; otherwise, serious numerical oscillation (spurious solutions) will occur. The implicit time integration scheme possesses a wider stability region than the explicit method; therefore, it is suitable for stiff problems. To investigate an efficient solution method for the stiff Stokes flow system, we apply first (backward Euler (BE)) and second order (trapezoidal method (TR) and trapezoidal rule—backward difference formula (TR-BDF2)) accurate implicit methods for the MIC solution scheme. The introduction of implicit time integration schemes results in nonlinear systems of equations. We utilize a Jacobian free Newton Krylov (JFNK) based Newton framework to solve the resulting nonlinear equations. In this work we also investigate two efficient implicit solution strategies to reduce the computational cost when solving stiff nonlinear systems. The two methods differ in how the advective term in the material transport evolution equation is treated. We refer to the method that employs Lagrangian update as “fully implicit” (Imp), whilst the method that employs Eulerian update is referred to as “semi-implicit” (SImp). Using a finite difference (FD) method, we have performed a series of numerical experiments which clarify the accuracy of solutions and trade-off between the computational cost associated with the nonlinear solver and time step size. In comparison with the general explicit Euler method, the second order accurate Imp methods reduce total computational cost successfully through the utilization of a large time step without sacrificing accuracy and stability. Moreover, the proposed SImp method is effective in reducing the computational cost associated with evaluating the nonlinear residual while obtaining a solution similar to the Imp method.

Reducing the computational footprint for real-time BCPNN learning.

The implementation of synaptic plasticity in neural simulation or neuromorphic hardware is usually very resource-intensive, often requiring a compromise between efficiency and flexibility. A versatile, but computationally-expensive plasticity mechanism is provided by the Bayesian Confidence Propagation Neural Network (BCPNN) paradigm. Building upon Bayesian statistics, and having clear links to biological plasticity processes, the BCPNN learning rule has been applied in many fields, ranging from data classification, associative memory, reward-based learning, probabilistic inference to cortical attractor memory networks. In the spike-based version of this learning rule the pre-, postsynaptic and coincident activity is traced in three low-pass-filtering stages, requiring a total of eight state variables, whose dynamics are typically simulated with the fixed step size Euler method. We derive analytic solutions allowing an efficient event-driven implementation of this learning rule. Further speedup is achieved by first rewriting the model which reduces the number of basic arithmetic operations per update to one half, and second by using look-up tables for the frequently calculated exponential decay. Ultimately, in a typical use case, the simulation using our approach is more than one order of magnitude faster than with the fixed step size Euler method. Aiming for a small memory footprint per BCPNN synapse, we also evaluate the use of fixed-point numbers for the state variables, and assess the number of bits required to achieve same or better accuracy than with the conventional explicit Euler method. All of this will allow a real-time simulation of a reduced cortex model based on BCPNN in high performance computing. More important, with the analytic solution at hand and due to the reduced memory bandwidth, the learning rule can be efficiently implemented in dedicated or existing digital neuromorphic hardware.

Dirichlet-to-Neumann semigroup acts as a magnifying glass

The first aim of this paper is to illustrate numerically that the Dirichlet-to-Neumann semigroup represented by P. Lax acts as a magnifying glass. In this perspective, we used the finite element method for discretizing of the correspondent boundary dynamical system using the implicit and explicit Euler schemes. We prove by using the Chernoff’s Theorem that the implicit and explicit Euler methods converge to the exact solution and we use the (P1)-finite elements to illustrate this convergence through a FreeFem++ implementation which provides a movie available online. In the Dirichlet-to-Neumann semigroup represented by P. Lax the conductivity $$\gamma $$ is the identity matrix $$I_n$$ , but for a different conductivity $$\gamma $$ , the authors of Cornean et al. (J Inverse Ill-posed Prob 12:111–134, 2006) supplied an estimation of the operator norm of the difference between the Dirichlet-to-Neumann operator $$\Lambda _\gamma $$ and $$\Lambda _1$$ , when $$\gamma =\beta I_n$$ and $$\beta =1$$ near the boundary $$\partial \Omega $$ (see Lemma 2.1). We will use this result to estimate the accuracy between the correspondent Dirichlet-to-Neumann semigroup and the Lax semigroup, for $$f\in H^{1/2}(\partial \Omega )$$ .

Digital Sliding-Mode Sensorless Control for Surface-Mounted PMSM

In this work, a sampled-data model for surface-mounted permanent magnet synchronous motors (SMPMSMs) is proposed based on the symplectic Euler method. Then, based on the obtained model, a digital sliding-mode controller is designed for rotor velocity output tracking. Moreover, for rotor position and velocity and load estimation, a digital observer is designed. A separation principle is introduced in order to verify that the proposed controller can accomplish its task with estimated signals. Open-loop simulations predict that the sampled-data model obtained via the symplectic Euler method performs better than a model obtained with the classical explicit Euler method. Moreover, closed-loop simulations are carried on where the proposed controller is compared with a sampled continuous-time controller, where the controller here designed yields to better results even when the sampling period is increased. Finally, the performance of the proposed controller is verified by experimental tests.

The significance of spatial reconstruction in finite volume methods for the shallow water equations

We study the significance of the spatial reconstruction when solving the one dimensional shallow water equations using a finite volume method. For that aim, we implement the explicit forward Euler method for temporal integration while the spatial discretization is performed by finite volume method. We compare the results of constant spatial reconstruction with those of linear spatial reconstruction. The numerical tests include the steady state of a lake at rest, the steady state of moving water and an unsteady state of dam break problem. It is shown that the spatial reconstruction has a significant role in the accuracy of the finite volume method. 1412 Noor Hidayat, Suhariningsih, Agus Suryanto and Sudi Mungkasi