Backward Euler Method Research Articles

Finite element methods for fractional-order diffusion problems with optimal convergence order

A convergence result is stated for the numerical solution of space-fractional diffusion problems. For the spatial discretization, an arbitrary family of finite elements can be used combined with the matrix transformation technique. The analysis covers the application of the implicit Euler method for time integration to ensure unconditional stability. The spatial convergence rate does not depend on the fractional power of the Laplacian operator. An efficient numerical implementation is developed avoiding the direct computation of matrix powers.

SOL-KiT—Fully implicit code for kinetic simulation of parallel electron transport in the tokamak Scrape-Off Layer

Here we present a new code for modelling electron kinetics in the tokamak Scrape-Off Layer (SOL). SOL-KiT (Scrape-Off Layer Kinetic Transport) is a fully implicit 1D code with kinetic (or fluid) electrons, fluid (or stationary) ions, and diffusive neutrals. The code is designed for fundamental exploration of non-local physics in the SOL and utilizes an arbitrary degree Legendre polynomial decomposition of the electron distribution function, treating both electron–ion and electron–atom collisions. We present a novel method for ensuring particle and energy conservation in inelastic and superelastic collisions, as well as the first full treatment of the logical boundary condition in the Legendre polynomial formalism. To our knowledge, SOL-KiT is the first fully implicit arbitrary degree harmonic kinetic code, offering a conservative and self-consistent approach to fluid–kinetic comparison with its integrated fluid electron mode. In this paper we give the model equations and their discretizations, as well as showing the results of a number of verification/benchmarking simulations. Program summaryProgram Title: SOL-KiTCPC Library link to program files:http://dx.doi.org/10.17632/7p44sdfnnp.1Licencing provisions: GNU GPLv3Programming language: Fortran 90Nature of problem:Fluid models of parallel transport in the Scrape-Off Layer (SOL) fail to account for the fact that the electron distribution function is often far from a Maxwellian, and kinetic effects have been linked to discrepancies between experiment and fluid modelling[1]. A kinetic treatment of electrons in the SOL requires detailed accounting of collisional processes, especially those with neutral particles, as well as a proper implementation of the logical boundary condition at the material surface[2]. Furthermore, the ability to identify differences between fluid and kinetic modelling using self-consistent comparison is desirable.Solution method: Electrons are modelled either as a fluid, or kinetically, maintaining self-consistency between models. All equations are solved using finite difference and the implicit Euler method, with fixed-point iteration. The kinetic approach is based on solving the Vlasov–Fokker–Planck–Boltzmann equation, decomposed in Legendre polynomials. Equations for the harmonics (or electron fluid equations) are solved alongside fluid equations for the ions, Ampère–Maxwell’s law for the electric field, as well as a diffusive–reactive Collisional-Radiative model for the evolution of hydrogenic atomic states. Each individual operator is built into a matrix, combined with all other operators, and the matrix equation arising from the Euler method is solved using the PETSc library, with MPI parallelization.Additional comments:This article presents the physical and numerical outline of the code, and is accompanied by html documentation, a small test suite based on benchmarking runs presented below, as well as instructions and means for compiling and executing SOL-KiT. Special focus in the article is given to the novel numerical and model aspects in the greater context of the developed software.

A Finite Element Splitting Method for a Convection-Diffusion Problem

Abstract For a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k / m {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

A Cyclic Plasticity Model with Martensite Transformation for S30408 and Its Finite Element Implementation

The ability of the constitutive model to simulate the ratcheting behavior of metastable austenitic stainless steel S30408 is significant to ensure the safety of the liquefied natural gas (LNG) semi-trailer tanks in the lightweight process of the inner containers. This is because the lightweight inner vessels often encounter cyclic stresses due to the road inertia loads together with high mean stresses due to internal pressures. In this study, we performed cryogenic uniaxial tension experiments and a series of ratcheting experiments to investigate the cyclic plasticity behavior of the metastable austenitic stainless steel S30408. Based on the Ohno-Wang II model, we proposed a new cyclic plasticity constitutive model with martensitic transformation, which relates the content of deformation-induced martensite with isotropic hardening and kinematic hardening. The ratcheting behaviors of S30408 were first simulated by the proposed model with the incremental loading method using MATLAB. The results showed that the model could reasonably predict the ratcheting behavior of S30408, and the evolution law of martensite content could well predict the content of deformation-induced martensite. Under the assumption of the von Mises yield criterion and normal plasticity flow rule, we developed a numerical algorithm of plastic strain with the proposed model to implement the finite element calculation of the model. Internal iteration in the numerical algorithm was implemented with the Euler backward method, which calculated the trial strain for each equilibrium iteration using the consistent tangent matrix. With a user subroutine, the proposed model was programmed into ANSYS for a user - executable version. By simulating the uniaxial ratcheting of a S30408 round bar, we found that the calculated results were in good agreement with the experimental results, which promises further applications in the design of structures, such as LNG semi-trailer tanks.

SIERE

Physics-based simulation methods for deformable objects suffer limitations due to the conflicting requirements that are placed on them. The work horse semi-implicit (SI) backward Euler method is very stable and inexpensive, but it is also a blunt instrument. It applies heavy damping, which depends on the timestep, to all solution modes and not just to high-frequency ones. As such, it makes simulations less lively, potentially missing important animation details. At the other end of the scale, exponential methods (exponential Rosenbrock Euler (ERE)) are known to deliver good approximations to all modes, but they get prohibitively expensive and less stable for very stiff material. In this article, we devise a hybrid, semi-implicit method called SIERE that allows the previous methods SI and ERE to each perform what they are good at. To do this, we employ at each timestep a partial spectral decomposition, which picks the lower, leading modes, applying ERE in the corresponding subspace. The rest is handled (i.e., effectively damped out) by SI. No solution of nonlinear algebraic equations is required throughout the algorithm. We show that the resulting method produces simulations that are visually as good as those of the exponential method at a computational price that does not increase with stiffness, while displaying stability and damping with respect to the high-frequency modes. Furthermore, the phenomenon of occasional divergence of SI is avoided.

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

AbstractIn this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P0, P0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.

A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

Abstract We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k m {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} . This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

SDFEM for singularly perturbed parabolic initial-boundary-value problems on equidistributed grids

In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler method to discretize the temporal derivative and the SDFEM scheme for the spatial derivatives. The proposed method is uniformly convergent with first-order in time and second-order in space.

Применение упрощенного неявного метода Эйлера для решения задач электрофизиологии

Рассматривается упрощенный неявный метод Эйлера как альтернатива явному методу Эйлера, являющемуся наиболее распространенным в области численного решения уравнений, описывающих электрическую активность нервных клеток и кардиоцитов. Многие модели электрофизиологии имеют высокую степень жесткости, так как описывают динамику процессов с существенно разными характерными временами: миллисекундная деполяризации предшествует значительно более медленной гиперполяризации при формировании потенциала действия в электровозбудимых клетках. Оценка степени жесткости в работе проводится по формуле, не требующей вычисления собственных значений матрицы Якоби системы ОДУ. Эффективность численных методов сравнивается на примере типичных представителей из классов детальных и концептуальных моделей возбудимых клеток: модели Ходжкина–Хаксли для нейронов и Алиева–Панфилова для кардиоцитов. Сравнение эффективности численных методов проведено с использованием распространенных в биомедицинских задачах видов норм. Исследовано влияние степени жесткости моделей на величину ускорения при использовании упрощенного неявного метода: выигрыш во времени при высокой степени жесткости зафиксирован только для модели Ходжкина–Хаксли. Обсуждаются целесообразность применения простых методов и методов высоких порядков точности для решения задач электрофизиологии, а также устойчивость методов. Обсуждение позволяет прояснить вопрос о причинах отказа от использования высокоточных методов в пользу простых при проведении практических расчетов. На примере модели Ходжкина–Хаксли c различными степенями жесткости вычислены производные решения высших порядков и обнаружены их значительные максимальные абсолютные значения. Последние входят в формулы констант аппроксимации и, следовательно, нивелируют малость множителя, зависящего от порядка точности. Этот факт не позволяет считать погрешности численного метода малыми. Проведенный на качественном уровне анализ устойчивости явного метода Эйлера позволяет оценить вид функции параметров модели для описания границы области устойчивости. Описание границы области устойчивости, как правило, используется при априорном принятии решения о выборе величины шага численного интегрирования.

A higher order scheme for singularly perturbed delay parabolic turning point problem

PurposeThe purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point.Design/methodology/approachThe authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme.FindingsThe proposed method has a convergence rate of order . Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor.Originality/valueA class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

This paper is concerned with the strong convergence of numerical methods for stochastic differential equations with a variable delay. Based on the concepts of stochastic C-stability and B-consistency, we first derive a fundamental strong convergence theorem for general one-step methods applied to nonlinear autonomous stochastic delay differential equations under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. These conditions admit that the diffusion coefficient is highly nonlinear. Then, we construct the split-step backward Euler method. It is verified that the method is stochastically C-stable and B-consistent of order 1/2 so that the method is strongly convergent of order 1/2 for the considered problems. Finally, the theoretical results are confirmed by numerical experiments.

An enhanced sequential fully implicit scheme for reservoir geomechanics

In this paper, it is proposed an enhanced sequential fully implicit (ESFI) algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit (SFI) algorithm is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy’s law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot’s theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method, and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy’s flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy’s flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poro-elastoplastic problems.

NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS USING IMPLICIT MILSTEIN METHOD

Stiff stochastic differential equations arise in many applications including in the area of biology. In this paper, we present numerical solution of stochastic differential equations representing the Malthus population model and SIS epidemic model, using the improved implicit Milstein method of order one proposed in [6]. The open source programming language SCILAB is used to perform the numerical simulations. Results show that the method is more accurate and stable compared to the implicit Euler method.

A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

A Finite Difference Method for Space Fractional Differential Equations with Variable Diffusivity Coefficient

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

Performance Analysis of Speed-Sensorless Induction Motor Drive Using Discrete Current-Error Based MRAS Estimators

In the literature on sensorless control of induction motors, many algorithms have been presented for rotor flux and speed estimation. However, all these algorithms have been developed in the continuous–time domain. The digital realization of the control systems, requires the implementation of those estimation methods in a discrete–time domain. The main goal of this article is comparison of the impact of different numerical integration methods, used in analogue emulation under the digital implementation of the control systems, to the operation of classical Model Reference Adaptive System; CC-based on two current models (MRASCC) speed estimator and its three modified versions developed for the extension of the estimator stability region. In this paper the generalized mathematical model of MRASCC estimator is proposed, which takes into account all known methods for the extension of the stability region of classical speed estimator of this type. After the short discussion of the discretization methods used for the microprocessor implementation of control algorithms the impact of different numerical integration methods on the stable operation range of the classical and modified MRASCC estimators is analyzed and validated in simulation and experimental tests. It is proved that Modified Euler discretization method is much more accurate than forward and backward Euler methods and gives almost as accurate results as Tustin method, however is much less complicated in practical realization.

Nonlinear Galerkin finite element methods for fourth-order Bi-flux diffusion model with nonlinear reaction term

A fourth-order diffusion model is presented with a nonlinear reaction term to simulate some special chemical and biological phenomenon. To obtain the solutions to those problems, the nonlinear Galerkin finite element method under the framework of the Hermite polynomial function for the spatial domain is utilized. The Euler backward difference method is used to solve the equation in the temporal domain. Subject to the Dirichlet and Navier boundary conditions, the numerical experiments for Bi-flux Fisher–Kolmogorov model present excellent convergence, accuracy and acceleration behavior. Also, the numerical solutions to the Bi-flux Gray–Scott model, subject to no flux boundary conditions, show excellent convergence, accuracy and symmetry.

Refactorization of the midpoint rule

An alternative formulation of the midpoint method is employed to analyze its advantages as an implicit second-order absolutely stable timestepping method. Legacy codes originally using the backward Euler method can be upgraded to this method by inserting a single line of new code. We show that the midpoint method, and a theta-like generalization, are B-stable. We outline two estimates of local truncation error that allow adaptive time-stepping.

Positional finite element formulation for two-dimensional analysis of elasto-plastic solids with contact applied to cold forming processes simulation

In this work, we develop a position-based finite element formulation for elasto-plastic solids under contact situation. The proposed positional formulation employs a total Lagrangian description and naturally considers geometric nonlinearities. The employed elasto-plastic model is derived from the dissipation inequality, using the thermodynamic conjugacy between the plastic strain rate and the so-called Mandel stress. The formulation is based on the Kroner–Lee decomposition, in which the deformation gradient is multiplicatively split into its elastic and plastic parts. We apply the backward Euler method to solve the plastic evolutions and von Mises yield criterion to define the elastic limit. The adopted kinematic hardening model is a finite strain generalization of the Armstrong–Frederick law, which uses the objective Jaumann derivative for the evolution equation and the concept of back stress tensor as an internal variable. For the elastic parcel of strains, we adopt a neo-Hookean constitutive law. With respect to the 2D application, plane strain and plane stress approximations are considered, where the latter is solved numerically by a local Newton–Raphson numerical procedure. Regarding the contact problem, a classical node-to-segment algorithm is applied, considering both frictionless and frictional cases, with the introduction of Lagrange multipliers in order to enforce contact constraints. Representative numerical examples are used to validate and show the possibilities of the proposed formulation in macroscale simulation of metal cold forming manufacturing processes.

A split scheme for the two-fluid seven-equation model with source terms

We develop a numerical solver based on the split scheme for the two-fluid seven-equation model. The solver contains two parts, that are the homogenous partial differential equations (PDEs) system solver and the ordinary differential equations (ODEs) system solver. The homogenous PDEs system solver employs the weak formulation of Roe scheme, and the general equations of state (EOS) are considered. The ODEs system solver employs implicit methods which are backward Euler method, trapezium rule method and TR-BDF2, because the relaxation terms lead to stiffness. The algebraic equations system deduced from the ODEs system is solved by Newton method. The solver is tested to assess two-phase Riemann problem, relaxation terms test problems, faucet flow problem and sedimentation problem. The influence of the number of cells, the performances of ODE solvers and the numerical diffusion are discussed. The solver performs well. But, the prediction of shock solution is not accurate. The numerical diffusion is influenced by the sound speed.