Implicit Euler Scheme Research Articles

HyPix: 1D physically based hydrological model with novel adaptive time-stepping management and smoothing dynamic criterion for controlling Newton–Raphson step

The newly developed open-source Hydrological Pixel model, HyPix, written in the fast and flexible Julia language, efficiently solves the mixed form of the Richardson–Richards’ equation (RRE). HyPix uses a cell-centred, finite-volume scheme for the spatial discretization, with an implicit Euler scheme for the temporal discretization, by using the weighted average inter-cell hydraulic conductivity. HyPix includes the following modules: (a)rainfall interception, (b)root water uptake with compensation algorithm and root growth, (c)soil evaporation, (d)ponding using a novel method for computing sorptivity, and (e)runoff. HyPix includes a wide range of top and boundary conditions (flux, pressure, free drainage). To control the Newton–Raphson iterations, HyPix incorporates a novel dynamic physical smoothing criterion, which improves not only the model performance but also its accuracy compared with using the traditional absolute convergence criterion. To control the time-step, the traditional physical time-step management based on changes in the soil water content was specifically designed to solve RRE based on soil water content. This work adapts the time-step management such that it is specifically designed to solve RRE based on soil water pressure without introducing further parameters. The novel time-step management also requires only one parameter and was found to be more efficient than the traditional time-step management. HyPix implements an option to solve the derivatives numerically, enabling the RRE to be modified and tested (e.g., the inter-cell hydraulic conductivity) by changing only a few lines of code. Numerically calculating derivatives was found to be as accurate as deriving the derivatives analytically, and only 10–25% slower.The well-established hydrological model HYDRUS was used to validate HyPix without the sink term. The HyPix results show good agreement to HYDRUS, validating the algorithms implemented in HyPix. Even for challenging conditions, HyPix can provide accurate and reliable results using the recommended standard options. Moreover, the algorithm developed in HyPix is more efficient than the one used in HYDRUS, particularly for coarse texture soils. The recommended options were also tested by running HyPix with sink term using field data.

Mittag-Leffler Euler [formula omitted]-differences for Caputo fractional-order systems

Exponential Euler differences have got rapid development recently for integer-order differential equations. But there are few papers focusing on this difference to fractional differential equations. This paper establishes the basic structure of Mittag-Leffler implicit Euler scheme for fractional differential equations by using the constant variation methods in fractional calculus. More critically, since the acquired difference forms belong to the scope of implicit Euler differences and then the fractional PECE algorithms are proposed to solve these implicit differences in numerical calculations effectively. Besides, a novel nonlocal difference operator is proposed and some relative properties are presented. In the end, the existence and boundedness of a unique globally exponentially stable solution of some nonlocal difference system with short-term memory are investigated. Some calculative examples and numerical simulations are given to illustrate the primary research findings.

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.

Analysis of a Mathematical Model Arising in Plant Disease Epidemiology

In this work we study from the mathematical and numerical point of view a problem arising in vector-borne plant diseases. The model is written as a nonlinear system composed of a parabolic partial differential equation for the vector abundance function and a first-order ordinary differential equation for the plant health function. An existence and uniqueness result is proved using backward finite differences, uniform estimates and passing to the limit. The regularity of the solution is also obtained. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. A discrete stability property and a main a priori error estimates result are proved using a discrete version of Gronwall’s lemma and some estimates on the different approaches. Finally, some numerical results, in one and two dimensions, are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

Numerical approximation of some poro-elastic problems with MGT-type dissipation mechanisms

In this work, we numerically analyze a porous elastic problem including several dissipation mechanisms of MGT type. The resulting variational problem is written in terms of the acceleration and the porosity speed. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved from which the linear convergence of the approximation is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation, the discrete energy decay and the behavior of the solution.

Semi-discrete Optimization Through Semi-discrete Optimal Transport: A Framework for Neural Architecture Search

In this paper, we introduce a theoretical framework for semi-discrete optimization using ideas from optimal transport. Our primary motivation is in the field of deep learning, and specifically in the task of neural architecture search. With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search [in the companion work (García-Trillos et al. in Traditional and accelerated gradient descent for neural architecture search, 2021); we show that algorithms inspired by our framework are competitive with contemporaneous methods]. We introduce a Riemannian like metric on the space of probability measures over a semi-discrete space \({\mathbb {R}}^d \times \mathcal {G}\) where \(\mathcal {G}\) is a finite weighted graph. With such Riemannian structure in hand, we derive formal expressions for the gradient flow of a relative entropy functional, as well as second-order dynamics for the optimization of said energy. Then, with the aim of providing a rigorous motivation for the gradient flow equations derived formally we also consider an iterative procedure known as minimizing movement scheme (i.e., Implicit Euler scheme, or JKO scheme) and apply it to the relative entropy with respect to a suitable cost function. For some specific choices of metric and cost, we rigorously show that the minimizing movement scheme of the relative entropy functional converges to the gradient flow process provided by the formal Riemannian structure. This flow coincides with a system of reaction–diffusion equations on \({\mathbb {R}}^d\).

On the numerical approximation of a problem involving a mixture of a MGT viscous material and an elastic solid

In this work, we analyze, from the numerical point of view, a problem including a mixture made of a MGT viscoelastic solid and an elastic solid. The corresponding variational problem is a linear system composed of two coupled hyperbolic equations written in terms of the acceleration of the first constituent and the velocity of the second one. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one-dimensional numerical simulations are shown to demonstrate the accuracy of the proposed approximations and the behaviour of the solution.

A numerical scheme to solve mixed parabolic-elliptic problem involving singular perturbation

In this article, a singularly perturbed mixed-type problem of parabolic-elliptic is considered on a rectangular domain. The solution of the problem possesses both boundary and interior layers which appears in the spatial variable. To approximate the time derivative, first the implicit Euler scheme and then the Crank–Nicolson scheme on uniform mesh in time direction is used. For approximating the spatial derivative, we use the central difference scheme in the first sub-domain and a hybrid scheme in the second sub-domain on Shishkin type meshes (standard Shishkin mesh and Bakhvalov–Shishkin mesh). We prove that both the numerical schemes converge uniformly with respect to the perturbation parameter and are of second-order accurate. Thomas algorithm is used to solve the tri-diagonal system. The numerical results and the error bounds are illustrated through few tables and graphs.

Marangoni convection within thermosolute and absorptive aqueous LiBr solution

The Marangoni convection during steam absorption into an aqueous lithium bromide (LiBr) solution is investigated numerically. In order to obtain a better understanding of the physical mechanism of thermal convection in the solution, the influence of surface tension on the absorbing solution is explored. For this purpose, this paper proposes the radial basis function (RBF) meshless method for numerical simulation of thermosolutal convection and vapor absorption in aqueous LiBr solution problems. The spatial terms of the governing equation are discretized using the RBF meshless method. The fractional step method is used for solving the continuity equations and momentum equations. The coupled energy and species equations are solved iteratively. The semi-algebraic system is solved using an implicit backward Euler scheme to maintain stability. The no-slip boundary conditions are imposed on solid wall surface. The surface tension gradient and the energy balance due the phase change are imposed on the interface of vapor and solution interface. The thermosolutal convection problem is presented to examine the accuracy of proposed method. An analytical solution is also presented for validating the vapor absorption problem results for zero Marangoni number (Ma). The Marangoni number used has a range of 0 ≤ Ma ≤ 1000. The increase in Marangoni number for low Prandtl number values does not significantly affect heat and mass transfer. However, increasing Ma for LiBr solution has a significant effect. It is also found that the velocity of convection flow can be improved by adding a solid object to the interface.

Long time behavior of discrete volume preserving mean curvature flows

Abstract In this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.

On the mixtures of MGT viscoelastic solids

<abstract><p>In this paper, we study, from both analytical and numerical points of view, a problem involving a mixture of two viscoelastic solids. An existence and uniqueness result is proved using the theory of linear semigroups. Exponential decay is shown for the one-dimensional case. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained and the linear convergence is derived under suitable regularity conditions. Finally, one- and two-dimensional numerical simulations are presented to demonstrate the convergence, the discrete energy decay and the behavior of the solution.</p></abstract>

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>

Numerical analysis of a problem in micropolar thermoviscoelasticity

<abstract><p>In this work, we study, from the numerical point of view, a dynamic thermoviscoelastic problem involving micropolar materials. The model leads to a linear system composed of parabolic partial differential equations for the displacements, the microrotation and the temperature. Its weak form is written as a linear system made of first-order variational equations, in terms of the velocity field, the microrotation speed and the temperature. Fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence is derived under some additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.</p></abstract>

On existence and numerical approximation in phase-lag thermoelasticity with two temperatures

<p style='text-indent:20px;'>In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerical analysis, we restrict ourselves to the case of second-order approximations of the heat flux and first-order approximations for the inductive temperature. First, variational formulation of the corresponding problem is derived and an energy decay property is proved. Then, a fully discrete scheme is introduced by using the finite element method for the approximation of the spatial variable and the implicit Euler scheme for the discretization of the time derivatives. A discrete stability</p>

Analysis of the thermoviscoelastic Timoshenko system with diffusion effect

This paper is concerned with a new Timoshenko beam model with thermal, mass diffusion and viscoelastic effects. First, by the C0-semigroup theory, we prove the well posedness of the considered problem with Dirichlet boundary conditions. An exponential decay is obtained by construction of the Lyapunov functional. Finally, a numerical study is done based on P 1 -finite element approximation for space discretization and implicit Euler scheme for time discretization, where we study the stability of the scheme, analysis of the error and obtained some numerical simulations.

Cortex tissue relaxation and slow to medium load rates dependency can be captured by a two-phase flow poroelastic model

This paper investigates the complex time-dependent behavior of cortex tissue, under adiabatic condition, using a two-phase flow poroelastic model. Motivated by experiments and Biot’s consolidation theory, we tackle time-dependent uniaxial loading, confined and unconfined, with various geometries and loading rates from 1μm/s to 100μm/s. The cortex tissue is modeled as the porous solid saturated by two immiscible fluids, with dynamic viscosities separated by four orders, resulting in two different characteristic times. These are respectively associated to interstitial fluid and glial cells. The partial differential equations system is discretized in space by the finite element method and in time by Euler-implicit scheme. The solution is computed using a monolithic scheme within the open-source computational framework FEniCS. The parameters calibration is based on Sobol sensitivity analysis, which divides them into two groups: the tissue specific group, whose parameters represent general properties, and sample specific group, whose parameters have greater variations. Our results show that the experimental curves can be reproduced without the need to resort to viscous solid effects, by adding an additional fluid phase. Through this process, we aim to present multiphase poromechanics as a promising way to a unified brain tissue modeling framework in a variety of settings.

Semi-implicit Hybrid Discrete $$\left( \text {H}^T_N\right) $$ Approximation of Thermal Radiative Transfer

The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing numerical solutions of the TRT equations accurately and efficiently is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P\(_N\) (spherical harmonics) or the S\(_N\) (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H\(^T_N\)) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P\(_N\) and S\(_N\), and indeed reduces to each of these approximations in various limits: H\(^1_N\) \(\equiv \) P\(_N\) and H\(^T_0\) \(\equiv \) S\(_T\). We prove that H\(^T_N\) results in a system of hyperbolic partial differential equations for all \(T\ge 1\) and \(N\ge 0\). Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop an asymptotic-preserving numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge–Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H\(^T_N\) ansatz and the numerical discretizations.

Longtime behavior of a semi-implicit scheme for Caginalp phase-field model

We present a semi-implicit scheme for the Caginalp phase-field model. The scheme is a combination of implicit Euler scheme together with Eyre’s decomposition. We prove the uniqueness of the numerical solution for small time-step and the unconditional stability of the scheme. We also show that the global attractors generated by the scheme converges to the global attractor of the continuous problem as the time step goes to zero.

Investigation of transient coupled conduction and radiation heat transfer in the linearly anisotropic scattering cylindrical medium by spectral collocation method

In this paper, a spectral collocation method (SCM) is developed to solve transient conduction and radiation heat transfer problems in infinitely long annular cylinders with participating medium. The participating medium is an absorbing, emitting and linearly anisotropic scattering medium. To avoid the complex calculation of ray tracing, the radiative transfer equation and the transient heat conduction equation are expressed in eulerian coordinates. In the solving process, for the radiative transfer equation, both spatial and angular domains are discretized by the SCM; for the transient heat conduction equation, the spatial domain is discretized by the SCM, and the time domain is discretized by the fully implicit Euler scheme. The spatial derivative term, angular derivative term and scattering source integral term are approximated by the high order spectral collocation scheme. Compared with available data in references, the SCM can achieve very good accuracy even using few nodes. Thus, the SCM is a very effective method to achieve high order accuracy for transient coupled conduction and radiation heat transfer in an anisotropic scattering cylindrical medium. Moreover, the effects of conduction-radiation parameter, optical thickness, and wall emissivity on the transient temperature distribution and energy flow rate have also been comprehensively investigated.

A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ( nge 3). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.