Backward Euler Method Research Articles

Accelerated parameter-uniform numerical method for singularly perturbed parabolic convection-diffusion problems with a large negative shift and integral boundary condition

The singularly perturbed parabolic convection–diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The implicit Euler method for the temporal direction and the exponentially fitted finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the method. The stability and uniform convergence analysis of the proposed method are studied. It is shown that the method is uniformly convergent with a convergence order of two in both temporal and spatial direction after Richardson extrapolation. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates. The proposed method provide more accurate results and a higher order of convergence than methods available in the literature.

Discrete Domain Design Scheme of Complex-Vector Current Controller for Induction Motor at Low Switching Frequency

The synchronous coordinates complex-vector proportional–integral (PI) controller is a high-performance current controller for vector-controlled ac motors, which is designed based on the continuous domain motor models. However, the application of digital signal processors in motor control necessitates the conversion of continuous domain models to discrete domain models. Moreover, the digital control itself has delay effects. Therefore, the original performance of continuous-domain-designed current controllers may be degraded. In order to improve the deteriorative performance, this article proposes corresponding delay compensation schemes for the forward Euler, backward Euler, and trapezoidal method discretized complex-vector PI controllers, respectively, which are analyzed based on discrete domain models. The root locus, frequency characteristics, and bandwidth are utilized to evaluate the performance of these three discretized controllers at the low switching frequency. Finally, the simulation and experimental results validate the analysis and the delay compensation methods proposed in this article.

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

AbstractIn this article, we have developed the mixed virtual element formulation for the nonlocal parabolic problem. A priori error estimates for the semi‐discrete and the fully‐discrete schemes are derived and analyzed. The spatial discretization is based on the mixed virtual element framework, and the backward Euler method is used for the time discretization. Using Brouwer's fixed point argument, we have proved the existence and uniqueness of a fully‐discrete scheme. A set of representative numerical examples investigates such theoretical results.

On stability in the maximum norm of difference scheme for nonlinear parabolic equation with nonlocal condition

We construct and analyze the backward Euler method for one nonlinear one-dimensional parabolic equation with nonlocal boundary condition. The main objective of this article is to investigate the stability and convergence of the difference scheme in the maximum norm. For this purpose, we use the M-matrices theory. We describe some new approach for the estimation of the error of solution and construct the majorant for it. Some conclusions and discussion of our approach are presented.

Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

In this paper, we propose four time marching schemes for the two-mode phase field crystal equation with the periodic boundary condition. The first- and second-order schemes are based on the stabilized semi-implicit approach with a combination of the backward Euler method, Crank–Nicolson method, and second-order backward differentiation formula, respectively. Based on the convex splitting method and third-order backward differentiation formula, a third-order scheme is developed. Different stabilized terms are added so that the energy stability of the proposed schemes can be guaranteed. Theoretically, the unique solvability and energy stability of the proposed schemes are rigorously proved. The error estimate of the first-order scheme is also given. Various numerical examples and simulations in the 2D and 3D cases are presented to demonstrate the accuracy, stability, and efficiency of the proposed schemes.

A NSFD method for the singularly perturbed Burgers-Huxley equation

This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and efficient numerical solution for this equation via the classical numerical methods. To overcome this challenge, a nonstandard finite difference (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary differential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniform convergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme.

Monte Carlo burnup oscillations for thorium-based EPR fuel

Advanced codes and computational resources have made it possible to model large geometries with finer spatial discretisation, which are required for detailed fuel analysis and licencing. Unfortunately, these detailed models exhibit instabilities during Monte Carlo burnup calculations. Several solutions for stabilising the combined transport and depletion calculations have been developed. This study focusses on the burnup instabilities that arise specifically for thorium-based fuels and employed existing methods to test the robustness and accuracy. The methods are evaluated in terms of burnup, isotopic masses and flux plots. The results proved that various methods can be employed to stabilise the oscillations, specifically for the proposed modification of the EPR using thorium. In particular, using the stochastic implicit Euler method and reducing the time step length (maximum 11.3 days) produced stable results. Thorium-based fuel appears to be particularly sensitive to the evolution of time step length as well as Pa233 concentration.

Numerical analysis of the Linearly implicit Euler method with truncated Wiener process for the stochastic SIR model

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIR model. For the real significance of the numerical analysis on stochastic SIR model, a linearly implicit Euler method with truncated Wiener process is introduced. The numerical positivity is obtained by the truncated Wiener process, which is the basis for the investigation of convergence and dynamical behavior. The numerical dynamical behavior is obtained by an exponential presentation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIR model.

First order strong approximation of Ait–Sahalia-type interest rate model with Poisson jumps

For Ait–Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypotheses, the considered model takes values in positive domain. It is shown that the TJABEM can preserve the domain of the underlying problem. Furthermore, the first-order convergence rate of the TJABEM is recovered with respect to a Lp-error criterion. Numerical experiments are finally given to illustrate the theoretical results.

Invariant measure of the backward Euler method for stochastic differential equations driven by α$$ \alpha $$‐stable process

The backward Euler method is employed to approximate the invariant measure of a class of stochastic differential equations (SDEs) driven by ‐stable processes. The existence and uniqueness of the numerical invariant measure are proved. Then the numerical invariant measure is shown to converge to the underlying invariant measure. Numerical examples are provided to demonstrate the theoretical results.

Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities

In this article, we provide the rigorous mathematical convergence proof both in space and time of the two dimensional Black Scholes equation with stochastic volatility. The spatial approximation of this three dimensional problem is performed using the finite volume method coupled with a fitted technique to tackle the degeneracy in the Black Scholes operator, while the temporal discretization is performed using implicit Euler method. We provide a mathematical rigorous convergence proof in space and time of the full discretized scheme. Numerical results are presented to validate our theoretical results.

Numerical treatment of singularly perturbed unsteady Burger-Huxley equation

This article deals with the numerical treatment of a singularly perturbed unsteady Burger-Huxley equation. The equation is linearized using the Newton-Raphson-Kantorovich approximation method. The resulting linear equation is discretized using the implicit Euler method and an exponential spline method for time and space variables, respectively. Richardson's extrapolation technique is employed to increase the accuracy in the temporal direction. The stability and uniform convergence of the proposed scheme are investigated. The scheme is shown uniformly convergent with the order of convergence O(τ + ℓ2) and O(τ2 + ℓ2) before and after Richardson extrapolation, respectively. Several test examples are considered to validate the applicability and efficiency of the scheme. It is observed that the proposed scheme provides more accurate results than the methods available in the literature.

Weak galerkin finite element method for the linear Schrodinger equation

The numerical technique for 2D time dependent linear Schrodinger equation is the subject of this work. The approximations are produced using the weak Galerkin finite element technique with continous and discrete FEM, on relay , using the backward Euler method in time. Using the elliptic projection operator, we provide L2 error speculation for continues and discretely weak Galerkin finite element.

Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

A study on lung cancer using nabla discrete fractional-order model

This study proposes a nabla discrete fractional-order system of differential equations to model lung cancer and its interactions with lung epithelial cells, mutated cells, oncogenes, tumor suppressor genes, immune cells, cytokines, growth factors, angiogenic factors, and extracellular matrix. The proposed model can help predict cancer growth, metastasis, and response to treatment. Analytical results show the system is stable with a unique solution, and the model predicts that the immune system responds to cancer cells but eventually becomes overpowered. The numerical analysis employed the forward and backward Euler method and demonstrated that changes in parameter values have significant effects on the steady-state solution. The findings show that the growth of lung epithelial cells or their interaction with immune cells can cause an increase in the number of lung cancer cells. Conversely, an increase in cell death or a reduction in the interaction between lung epithelial cells and immune cells can decrease the number of lung cancer cells. The study highlights the usefulness of the nabla discrete fractional model in studying lung cancer dynamics.

Бикомпактные схемы для решения одногрупповой системы уравнений квазидиффузии совместно с уравнением энергии

High-order bicompact schemes are proposed for one of the stages of the HOLO algorithm for solving the radiative transport equation and implemented. The system of quasi-diffusion equations is solved together with the energy equation. The schemes have a fourth-order approximation in space on a two-point stencil, integration over time is carried out using the diagonal-implicit Runge–Kutta method of the third order of approximation, each stage of the method can be represented as an implicit Euler method. It is proposed to organize an iterative process to find a solution, since the coefficients of the system are non-linear functions of temperature. The schemes are used for a number of analytical tests, the convergence in time and space with the declared orders is shown.

Graph Approximating SM Discretization

Modern practical sliding mode (SM) control (SMC) involves discrete noisy sampling. Standard SMC discretizations feature the chattering effect, while implicit Euler methods need significant model knowledge, constant sampling steps and are difficult in application for higher relative degrees and multiple discontinuities. The proposed universal discretization method is based on the graph approximation of Filippov inclusions. It guaranties convergence to the continuous-time solutions and is computationally-simple. For the first time we demonstrate low-chattering discretization in the presence of large noises and multiple discontinuities.

Discrete-time Sliding Mode Control with Adaptive Reaching Law via Implicit Euler Method

Discrete-time Sliding Mode Control with Adaptive Reaching Law via Implicit Euler Method

The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.

Nonpolynomial Spline Method for Singularly Perturbed Time‐Dependent Parabolic Problem with Two Small Parameters

This study deals with the numerical solution of parabolic convection‐diffusion problems involving two small positive parameters and arising in modeling of hydrodynamics. To approximate the solution, the backward Euler method for time stepping and fitted trigonometric‐spline scheme for spatial discretization are considered on uniform meshes. The resulting scheme is shown to be uniformly convergent and its rate of convergence is one in the time variable and two in the space variable. The accuracy and rate of convergence are enhanced by using the Richardson extrapolation. To support the theoretically shown convergence analysis, we have taken some numerical examples and compared the absolute maximum error of the current method with some methods existing in the literature.