Explicit Euler Scheme Research Articles

Fast simulation strategy for capacitively-coupled plasmas based on fluid model

Fluid simulations are widely used in optimizing the reactor geometry and improving the performance of capacitively coupled plasma (CCP) sources in industry, so high computation speed is very important. In this work, a fast method for CCP fluid simulation based on the framework of Multi-physics Analysis of Plasma Sources (MAPS) is developed, which includes a multi-time-step explicit upwind scheme to solve electron fluid equations, a semi-implicit scheme and an iterative method with in-phase initial value to solve Poisson's equation, an explicit upwind scheme with limited artificial diffusion to solve heavy particle fluid equations, and an acceleration method based on fluid equation modification to reduce the periods required to reach equilibrium. In order to prove the validity and efficiency of the newly developed method, benchmarking against COMSOL and comparison with experimental data have been performed in argon discharges on the Gaseous Electronics Conference (GEC) reactor. Besides, the performance of each acceleration method is tested, and the results indicated that the multi-time-step explicit Euler scheme can effectively decline the computational burden in the bulk plasma and reduce the time cost on the electron fluid equations by half. The in-phase initial value method can greatly decrease the iteration times required to solve linear equations and lower the computational time of Poisson's equation by 77 %. The acceleration method based on equation modification can reduce the periods required to reach equilibrium by two-thirds.

An explicit Euler scheme for generalized [formula omitted]-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises

The current paper is devoted to the numerical integration of generalized n-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises. We submitted the Euler integrator for the integral form of this type of differential equation. Convergence analysis shows that the proposed scheme order of strong superconvergence is 1.0 when diffusion terms, in general form gl2(t,ρ), l=1,2, satisfies gl2(t,t)=0. While, the strong convergence order of the scheme is 1/2 if gl2(t,t)≠0. Especially, for convolution diffusion terms gl2(t,ρ)=gl2(t−ρ) our scheme has strong superconvergence order 1.0 when gl2(0)=0. And scheme strongly convergent with order 1/2 if gl2(0)≠0. Finally, four problems are considered to exhibit the efficiency and compatibility of the numerical integrator.

Analytical and numerical study of diffusion propelled surface growth phenomena

To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

Mean-field modelling of precipitation kinetics with a Fokker-Planck equation in the Lifshitz–Slyozov–Wagner space

Mean-field theory has been widely applied in modelling the kinetics of microstructure evolution. Considering a free energy contribution by microstructure entropy, a Fokker-Planck equation for precipitation kinetics is variationally derived from thermodynamic principles. Using dimensionless transformation into the Lifshitz-Slyozov-Wagner space, the Fokker-Planck equation is effectively solved within a fixed radius interval and an explicit Euler scheme, which is computationally advantageous. Through comparisons with the Ni-12.5 Al at. % alloy, it is shown that the Fokker-Planck equation can represent the typical stages of precipitation kinetics, i.e., nucleation, growth, and coarsening.

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

The modeling of physical and technological processes often involves solving stiff initial value problems. In most cases, their exact solution is difficult to find, while numerical schemes sometimes fail to produce a sufficiently accurate solution in acceptable computation time. Moreover, for some classes of problems, numerical solution schemes are unsuitable because of their insufficient stability. This paper deals with numerical methods based on solution continuation with respect to arguments of various types that make it possible to enhance the stability of explicit numerical schemes. Most frequently, the used best argument is hardly applicable to problems in which the integral curves grow at a superpower or nearly exponential rate. Previously, the authors proposed a modification of the best argument that alleviates these disadvantages. In the present paper, we estimate the domain of absolute stability of the explicit Euler scheme as applied to problems transformed to a modified best argument of special form and refine the proof of a similar estimate for initial value problems transformed to the best argument. A test initial value problem is used to verify the resulting theoretical estimates and to analyze the application of the modified best argument of solution continuation.

High-order Adams Network (HIAN) for image dehazing

Convolutional Neural Networks (CNN) are widely used in image dehazing. However, existing network frameworks are built based on manual design from practical experience, lacking interpretable result or theoretical guidelines. Recently, residual networks are regarded as the explicit Euler forward approximation of the ODE (Ordinary Differential Equation), and several ODE-inspired networks are proposed based on the low-order explicit Euler schemes. However, on the issues of system stability and training convergence, high-order Implicit Adams Predictor–Corrector (IAPC) methods have proven to be better than low-order explicit Euler methods. Hence, we extend the IAPC method to the High-order Implicit Adams Network (HIAN). To do so, we design a series of Implicit Adams Predictor–Corrector Blocks (IABs) based on the high-order IAPC methods, all of which give better stability and accuracy than the ones designed using the low-order Euler methods. Given that, we further propose the Implicit Adams Predictor–Corrector Module (IAM) by combining the Non-local Sparse Attention (NSA) and Attention Feature Fusion (AFF) with stacked IABs where the NSA explores the mutual-correlation among intermediate features with low computation cost via a sparse constraint, while the AFF fuses intermediate features by reweighting the features from stacked IABs adaptively. Moreover, because manual network design with IABs limits dehazing performance, the Neural Architecture Search (NAS) is used to find an optimal architecture automatically. This resulting design not only is interpretable for image dehazing but also provides a reliable guideline on future network designs. The experiments demonstrate that the proposed method outperforms most existing methods on both synthetic and real images.

On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes

We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.

A discrete unified gas-kinetic scheme for multi-species rarefied flows

A discrete unified gas kinetic scheme (DUGKS) is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame (AAP) kinetic model. Although the species collision operator in the AAP model conserves fully the mass, momentum, and energy for the mixture, it does not conserve the momentum and energy for each species due to the inter-species collisions. In this work, the species collision operator is decomposed into two parts: one part is fully conservative for the species and the other represents the excess part. With this decomposition, the kinetic equation is solved using the Strang-splitting method, in which the excess part of the collision operator is treated as a source, while the kinetic equation with the species conservative part is solved by the standard DUGKS. Particularly, the time integration of the source term is realized by either explicit or implicit Euler scheme. By this means, it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules, while the previous DUGKS [Zhang Y, Zhu L, Wang R et al, Phys Rev E 97(5):053306, 2018] of binary gases was only designed for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure under different Mach numbers and molar concentrations, the Couette flow under different mass ratios, and the pressure-driven Poiseuille flow in different flow regimes. The results are compared with those from other reliable numerical methods based on different models. And the influence of molecular model on the flow characteristics is studied. The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.

Numerical analysis of a Caginalp phase-field system in type III heat conduction

In this work, we study, from the numerical point of view, the Caginalp phase-field problem arising in type III thermoelasticity. The thermal problem is written, as a coupled nonlinear system, in terms of the order parameter and the thermal displacements. Then, a fully discrete approximation is proposed by using the finite element method and a combination of both implicit and explicit Euler schemes. A priori error estimates are proved. Finally, one-dimensional numerical simulations are performed to demonstrate the convergence of the approximations and the behavior of the discrete energy.

Stability for discrete time waveform relaxation methods based on Euler schemes

<abstract><p>Stability properties of discrete time waveform relaxation (DWR) methods based on Euler schemes are analyzed by applying them to two dissipative systems. Some sufficient conditions for stability of the considered methods are obtained; at the same time two examples of instability are given. To investigate the influence of the splitting functions and underlying numerical methods on stability of DWR methods, DWR methods based on different splittings and different numerical schemes are considered. The obtained results show that the stabilities of waveform relaxation methods based on an implicit Euler scheme are better than those based on explicit Euler scheme, and that the stabilities of waveform relaxation methods based on the classical splittings such as Gauss-Jacobi and Gauss-Seidel splittings are worse than those based on the eigenvalue splitting presented in this paper. Finally, numerical examples that confirm the theoretical results are presented.</p></abstract>

Stability of Euler Methods for Fuzzy Differential Equation

The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ.

Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

We introduce and investigate a notion of multivalued lambda -dissipative probability vector field (MPVF) in the Wasserstein space mathcal {P}_2({textsf {X} }) of Borel probability measures on a Hilbert space {textsf {X} }. Taking inspiration from the theories of dissipative operators in Hilbert spaces and of Wasserstein gradient flows for geodesically convex functionals, we study local and global well posedness of evolution equations driven by dissipative MPVFs. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove novel convergence results with optimal error estimates under an abstract stability condition, which do not rely on compactness arguments and also hold when {textsf {X} } has infinite dimension. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the Bénilan notion of integral solutions to dissipative evolutions in Banach spaces. Existence, uniqueness and stability of EVI solutions are then obtained under quite general assumptions, leading to the generation of a semigroup of nonlinear contractions.

On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation

From the classical explicit Euler scheme of first order, nonstandard finite difference (NSFD) schemes were envisioned to mimic the essential properties of the governing differential equation model for every time step-size. In the context of compartmental epidemiological models, these properties are generally concerned with the positivity of subpopulations, conservation laws (dynamics of the total population), and stability. However, for autonomous systems, the symmetry (self-adjoint) condition is not preserved. Compartmental epidemiological models are Poisson systems, so methods from geometric numerical integration should be applicable. It is found that symmetrization of NSFD schemes does not respect positivity for every step-size, though other characteristics are maintained and second order is reached. Composition through the Lie formalism can then be applied to obtain higher-order schemes. This is a more efficient and consistent alternative to Richardson's extrapolation, which has often been used to go beyond order one.

On the randomized Euler schemes for ODEs under inexact information

We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which the right-hand side functions satisfy Lipschitz condition globally or only locally. Moreover, we assume that only inexact discrete information, corrupted by some noise, about the right-hand side function is available. Optimality and stability of explicit and implicit randomized Euler algorithms are also investigated. Finally, we report the results of numerical experiments which support our theoretical conclusions.

Corrected finite element methods on unfitted meshes for Stokes moving interface problem

We propose an explicit in time corrected finite element method for a problem with a moving elastic interface in Stokes flow, using unfitted meshes and standard finite element approximation spaces. Piecewise polynomial correction functions are computed to accurately capture the discontinuities of the solution resulting from the elastic force applied to the fluid and are used to correct the load vector. The method is proven to be optimal for the steady state problem. Additionally, we propose a trigonometric interpolant to approximate and evolve the interface and explicit Euler scheme to advance in time. Numerical examples demonstrate the third and second order accuracy of the method for the velocity and pressure, respectively, and the stability of the numerical scheme.

Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise

<p style='text-indent:20px;'>This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the solution to the sNLS equation. In the paper we first carry out a detailed analysis of the properties of the solution which lays down a theoretical foundation and guidance for numerical analysis, we then present a family of three-parameters fully discrete finite element methods which differ mainly in their time discretizations and contains many well-known schemes (such as the explicit and implicit Euler schemes and the Crank-Nicolson scheme) with different combinations of time discetization strategies. The prototypical <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>-schemes are analyzed in detail and various stability properties are established for its numerical solution. An extensive numerical study and performance comparison are also presented for the proposed fully discrete finite element schemes.</p>

Magnetic field simulations using explicit time integration with higher order schemes

PurposeA transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the resulting system of differential algebraic equations is reformulated into a system of ordinary differential equations (ODE). The ODE system is integrated in time by using explicit time integration schemes. The purpose of this paper is to investigate explicit time integration for eddy current problems with respect to the performance of the first-order explicit Euler scheme and the Runge-Kutta-Chebyshev (RKC) method of higher order.Design/methodology/approachThe ODE system is integrated in time using the explicit Euler scheme, which is conditionally stable by a maximum time step size. To overcome this limit, an explicit multistage RKC time integration method of higher order is used to enlarge the maximum stable time step size. Both time integration methods are compared regarding the overall computational effort.FindingsThe numerical simulations show that a finer spatial discretization forces smaller time step sizes. In comparison to the explicit Euler time integration scheme, the multistage RKC method provides larger stable time step sizes to diminish the overall computation time.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

Meshless simulation of a lid-driven cavity problem with a non-Newtonian fluid

The purpose of the present paper is to solve the lid-driven cavity problem for a non-Newtonian power-law shear thinning and shear thickening fluid by a meshless method. Results are presented for Re=100 and Re=1000, where different levels of shear thinning and thickening are considered. Furthermore, the lid-driven cavity case is made geometrically more complex by adding several circular-shaped obstacles in the geometry. The Navier-Stokes equations are solved with the local meshless diffuse approximate method. The weighted least squares approximation is structured by using the second-order polynomial basis vector and the Gaussian weight function. The explicit Euler scheme is used to perform the artificial time stepping. The non-incremental pressure correction scheme is used to couple the pressure and the velocity fields. Results are presented in terms of viscosity contours, stream function, velocity vectors, mid-plane velocity and viscosity profiles for the steady state. The numerical method is verified by a comparison with the results obtained from the literature, which are computed with the least squares finite element formulation. Furthermore, an investigation of the node density and node distribution type is performed. A near perfect match with the reference solution and between the structured and unstructured node distribution is found.

Fractional Damping Through Restricted Calculus of Variations

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton’s principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton’s principle relies on including fractional derivatives to the state space, the doubling of curves (which implies an extra mirror system) and the restriction of the class of varied curves. We will obtain the correct dynamics and will show rigorously that the extra mirror dynamics is nothing but the principal one in reversed time; thus, the restricted Hamilton’s principle is not adding extra physics to the original system. The price to pay, on the other hand, is that the fractional damped dynamics is only a sufficient condition for the extremals of the action. In addition, we proceed to discretise the new principle. This discretisation provides a set of numerical integrators for the continuous dynamics that we denote Fractional Variational Integrators (FVIs). The discrete dynamics is obtained upon the same ingredients, say doubling of discrete curves and restriction of the discrete variations. We display the performance of the FVIs, which have local truncation order 1, in two examples. As other integrators with variational origin, for instance those generated by the discrete Lagrange–d’Alembert principle, they show a superior performance tracking the dissipative energy, in opposition to direct (order 1) discretisations of the dissipative equations, such as explicit and implicit Euler schemes.

Finite element analysis of bone remodelling with piezoelectric effects using an open-source framework

Bone tissue exhibits piezoelectric properties and thus is capable of transforming mechanical stress into electrical potential. Piezoelectricity has been shown to play a vital role in bone adaptation and remodelling processes. Therefore, to better understand the interplay between mechanical and electrical stimulation during these processes, strain-adaptive bone remodelling models without and with considering the piezoelectric effect were simulated using the Python-based open-source software framework. To discretise numerical attributes, the finite element method (FEM) was used for the spatial variables and an explicit Euler scheme for the temporal derivatives. The predicted bone apparent density distributions were qualitatively and quantitatively evaluated against the radiographic scan of a human proximal femur and the bone apparent density calculated using a bone mineral density (BMD) calibration phantom, respectively. Additionally, the effect of the initial bone density on the resulting predicted density distribution was investigated globally and locally. The simulation results showed that the electrically stimulated bone surface enhanced bone deposition and these are in good agreement with previous findings from the literature. Moreover, mechanical stimuli due to daily physical activities could be supported by therapeutic electrical stimulation to reduce bone loss in case of physical impairment or osteoporosis. The bone remodelling algorithm implemented using an open-source software framework facilitates easy accessibility and reproducibility of finite element analysis made.