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Journal articles on the topic 'Backward Euler Method'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Soomro, Paras, Israr Ahmed, Faraz Ahmed Soomro, and Darshan Mal. "Numerical Simulation Model of the Infectious Diseases by Comparing Backward Euler Method and Adams-Bash forth 2-Step Method." VFAST Transactions on Mathematics 12, no. 1 (2024): 402–14. http://dx.doi.org/10.21015/vtm.v12i1.1881.

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In this work, the Backward Euler technique and the Adams-Bashforth 2-step method—two numerical approaches for solving the SIR model of epidemiology are compared for performance. An essential resource for comprehending the transmission of infectious illnesses like COVID-19 in the SIR model. While the explicit Adams-Bash forth 2-step approach is well known for its computing efficiency, the implicit Backward Euler method is noted for its stability. The study evaluates the accuracy, strength, and computing cost of the two approaches to determine which approach is best for simulating the spread of

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2

Nguyen, Anh Tan. "Comparative Study of Discretization Methods for Non-Ideal Proportional-Resonant Controllers in Voltage Regulation of Three-Phase Four-Wire Converters with Vehicle-to-Home Mode." World Electric Vehicle Journal 16, no. 6 (2025): 335. https://doi.org/10.3390/wevj16060335.

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Vehicle-to-home (V2H) technology enables electric vehicles (EVs) to supply power to residential loads, offering enhanced energy self-sufficiency and backup capabilities. Accurate voltage regulation is essential in such systems, especially under nonlinear and time-varying load conditions. The control method for three-phase four-wire (3P4W) converters plays a vital role in addressing these challenges. In the control configuration of such systems, the non-ideal proportional-resonant (PR) controller stands out due to its ability to reject periodic disturbances. However, the comprehensive study on

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3

Peskin, Charles S., and Tamar Schlick. "Molecular dynamics by the Backward-Euler method." Communications on Pure and Applied Mathematics 42, no. 7 (1989): 1001–31. http://dx.doi.org/10.1002/cpa.3160420706.

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4

Botchev, Mikhail Aleksandrovich, and Victor Timofeevich Zhukov. "Exponential Euler and backward Euler methods for nonlinear heat conduction problems." Keldysh Institute Preprints, no. 69 (2023): 1–16. http://dx.doi.org/10.20948/prepr-2023-69.

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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

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5

Avila-Becerril, Sofia, Alejandro Garcés-Ruiz, and Gerardo Espinosa-Pérez. "Discretization of Models for Electrical Microgrids Preserving Passivity Properties." Memorias del Congreso Nacional de Control Automático 7, no. 1 (2024): 173–78. https://doi.org/10.58571/cnca.amca.2024.030.

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This paper presents a Hamiltonian model of a direct current microgrid consisting of rectifiers connected in parallel in closed-loop with a Droop-type power-sharing algorithm. The main contribution is to demonstrate that a discretization using the backward Euler method preserves the microgrid’s Hamiltonian structure and passivity properties with a simple implementation of the method. The results are numerically evaluated by comparing the backward and forward Euler methods.

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6

Yu, Zhanhua, and Mingzhu Liu. "Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations." Discrete Dynamics in Nature and Society 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/217672.

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We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.

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7

Sumithra, B., and A. Tamilselvan. "Numerical solution of stiff system by backward Euler method." Applied Mathematical Sciences 9 (2015): 3303–11. http://dx.doi.org/10.12988/ams.2015.54317.

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8

Tian, Zhikun, Yanping Chen, and Jianyun Wang. "Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation." Mathematics 11, no. 14 (2023): 3127. http://dx.doi.org/10.3390/math11143127.

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We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the a

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9

Thomée, Vidar. "A Finite Element Splitting Method for a Convection-Diffusion Problem." Computational Methods in Applied Mathematics 20, no. 4 (2020): 717–25. http://dx.doi.org/10.1515/cmam-2020-0128.

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AbstractFor 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\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

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10

Slodička, Marián, and Edita Janíková. "Convergence of the backward Euler method for type-II superconductors." Journal of Mathematical Analysis and Applications 342, no. 2 (2008): 1026–37. http://dx.doi.org/10.1016/j.jmaa.2007.12.043.

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11

Peskin, Charles S. "Analysis of the backward-euler/langevin method for molecular dynamics." Communications on Pure and Applied Mathematics 43, no. 5 (1990): 599–645. http://dx.doi.org/10.1002/cpa.3160430503.

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12

Bock, Hans Georg, Jürgen Gutekunst, Andreas Potschka, and María Elena Suaréz Garcés. "A Flow Perspective on Nonlinear Least-Squares Problems." Vietnam Journal of Mathematics 48, no. 4 (2020): 987–1003. http://dx.doi.org/10.1007/s10013-020-00441-z.

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AbstractJust as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newt

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13

娜扎开提, ·阿迪力. "Numerical Analysis of Backward Euler-Galerkin Method for Water Contamination Problem." Advances in Applied Mathematics 06, no. 09 (2017): 1146–50. http://dx.doi.org/10.12677/aam.2017.69139.

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14

Jebens, Stefan, Oswald Knoth, and Rüdiger Weiner. "Explicit Two-Step Peer Methods for the Compressible Euler Equations." Monthly Weather Review 137, no. 7 (2009): 2380–92. http://dx.doi.org/10.1175/2008mwr2671.1.

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A new time-splitting method for the integration of the compressible Euler equations is presented. It is based on a two-step peer method, which is a general linear method with second-order accuracy in every stage. The scheme uses a computationally very efficient forward–backward scheme for the integration of the high-frequency acoustic modes. With this splitting approach it is possible to stably integrate the compressible Euler equations without any artificial damping. The peer method is tested with the dry Euler equations and a comparison with the common split-explicit second-order three-stage

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15

von der Weth, Axel, Daniela Piccioni Koch, Frederik Arbeiter, et al. "Numerical Solution Strategies in Permeation Processes." Defect and Diffusion Forum 413 (December 17, 2021): 29–46. http://dx.doi.org/10.4028/www.scientific.net/ddf.413.29.

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In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM for

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16

Guo, Qian, and Xueyin Tao. "Almost Sure andLpConvergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/390418.

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The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven inLp-sense. Almost sure convergence is derived from theLpconvergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.

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17

Milošević, Marija. "Divergence of the backward Euler method for ordinary stochastic differential equations." Numerical Algorithms 82, no. 4 (2019): 1395–407. http://dx.doi.org/10.1007/s11075-019-00661-6.

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18

Öner, Tarkan, and Osman Raşit Işık. "Accelerating convergence for Backward-Euler method by adding time relaxation term." Journal of Modern Methods in Numerical Mathematics 5, no. 2 (2014): 1. http://dx.doi.org/10.20454/jmmnm.2014.710.

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19

Aziz, Dalal Ismael, and Ahmed J. Hussein. "Weak galerkin finite element method for the linear Schrodinger equation." Journal of Interdisciplinary Mathematics 26, no. 5 (2023): 919–23. http://dx.doi.org/10.47974/jim-1529.

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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.

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20

Qiu, Meilan, Liquan Mei, and Dewang Li. "Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations." Advances in Mathematical Physics 2017 (2017): 1–20. http://dx.doi.org/10.1155/2017/4961797.

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We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter (0<α<1) or second-order central difference method for (1<α<2), combined with local discontinuous Galerkin method to approximate

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21

Nigro, Alessandra. "BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows." Fluids 8, no. 11 (2023): 304. http://dx.doi.org/10.3390/fluids8110304.

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In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and

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22

Guo, Qian, and Xueyin Tao. "The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/621359.

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Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

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23

Erath, Christoph, and Robert Schorr. "Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems." Computational Methods in Applied Mathematics 20, no. 2 (2020): 251–72. http://dx.doi.org/10.1515/cmam-2018-0253.

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AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate sinc

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24

Hajaji, Abdelmajid El, Abdelhafid Serghini, Said Melliani, Jalila El Ghordaf, and Khalid Hilal. "A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems." Tatra Mountains Mathematical Publications 80, no. 3 (2021): 15–34. http://dx.doi.org/10.2478/tmmp-2021-0029.

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Abstract In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the backward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of order

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25

Sofiane, Dehilis, Bouziani Abdelfatah, and Bensaid Souad. "A MODIFIED BACKWARD EULER SCHEME FOR THE DIFFUSION EQUATION SUBJECT TO NONLINEAR NONLOCAL BOUNDARY CONDITIONS." Eurasian Journal of Mathematical and Computer Applications 9, no. 3 (2021): 26–38. http://dx.doi.org/10.32523/2306-6172-2021-9-3-26-38.

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In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differ- entiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

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26

Wang, Xiaojie, and Siqing Gan. "The improved split-step backward Euler method for stochastic differential delay equations." International Journal of Computer Mathematics 88, no. 11 (2011): 2359–78. http://dx.doi.org/10.1080/00207160.2010.538388.

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27

Sun, Zhuo, Huaiming Zhu, and Haotian Xu. "Distributed Newton method for time-varying convex optimization with backward Euler prediction." AIMS Mathematics 9, no. 10 (2024): 27272–92. http://dx.doi.org/10.3934/math.20241325.

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<p>We investigated the challenge of unconstrained distributed optimization with a time-varying objective function, employing a prediction-correction approach. Our method introduced a backward Euler prediction step that used the differential information from consecutive moments to forecast the trajectory's future direction. This predicted value was then refined through an iterative correction process. Our analysis and experimental results demonstrated that this approach effectively addresses the optimization problem without requiring the computation of the Hessian matrix's inverse.</p&

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28

Zhang, Haomin, Siqing Gan, and Lin Hu. "The split-step backward Euler method for linear stochastic delay differential equations." Journal of Computational and Applied Mathematics 225, no. 2 (2009): 558–68. http://dx.doi.org/10.1016/j.cam.2008.08.032.

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29

Palencia, C. "Backward Euler method for abstract time-dependent parabolic equations with variable domains." Numerische Mathematik 82, no. 3 (1999): 471–90. http://dx.doi.org/10.1007/s002110050427.

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30

Songsanga, Danuruj, and Parinya Sa Ngiamsunthorn. "Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels." AIMS Mathematics 7, no. 8 (2022): 15002–28. http://dx.doi.org/10.3934/math.2022822.

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<abstract><p>We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular rule (forward Euler method), implicit product integration rectangular rule (backward Euler method), implicit product integration trapezoidal rule and Adam-type predictor-corrector method. In addition, the error estimation and stability for all four presented schemes are analyzed. To demonstrate the accuracy and effectiveness of the p

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31

Li, Rong Hua, Li Yang, and Jia Wei Li. "Split-Step Backward Euler Method for Stochastic Delay Hopfield Neural Networks with Markovian Switching." Applied Mechanics and Materials 58-60 (June 2011): 1390–95. http://dx.doi.org/10.4028/www.scientific.net/amm.58-60.1390.

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In this paper, split-step backward Euler method for stochastic delay Hopfield neural networks with Markovian switching is considered. The main aim of this paper is to show that the numerical approximation solution is convergent to the true solution with order. The conditions under which the numerical solution is exponentially stable in mean square are given. An example is provided for illustration.

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32

Mauser, Norbert J., and Yong Zhang. "Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System." Communications in Computational Physics 16, no. 3 (2014): 764–80. http://dx.doi.org/10.4208/cicp.110813.140314a.

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AbstractWe study the computation of ground states and time dependent solutions of the Schrödinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two space dimensions). On a disc-shaped domain, we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion in θ, and propose a second order finite difference scheme to solve ther-variable ODEs of the Fourier coefficients. The Poisson potential can be solved within(MNlogN) arithmetic operations whereM,Nare the number of grid points inr-direction and the Fourier bases. Combined with the

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33

Wang, Jianyun, and Yunqing Huang. "Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (2017): 671–88. http://dx.doi.org/10.4208/nmtma.2017.y16008.

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AbstractThis paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimalL2error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

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34

Ahmad, Takiyuddin Abdullah, Mahdaliza Idrus Sevia, and Md Ayob Shahrin. "Model predictive control using Euler method for switchedbattery boost-multilevel inverter." International Journal of Power Electronics and Drive Systems 14, no. 3 (2023): 1497~1508. https://doi.org/10.11591/ijpeds.v14.i3.pp 1497-1508.

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This paper presents the model predictive control (MPC) design using the backward Euler method for an 11-level switched-battery boost multilevel inverter (SBBMLI). The SBBMLI was proposed as the cost-effective solution for the communication power line in a high-speed rail (HSR) system. Initially, a generalized SBBMLI problem formulation was performed, and an open-loop simulation based on voltage and current mode controls was conducted. The finite control set-model predictive control (FCS-MPC) ability to track the reference sinusoidal output with low total harmonic distortion (THD) was then asse

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35

Kovács, Mihály, Stig Larsson, and Fredrik Lindgren. "On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation." Journal of Applied Probability 52, no. 2 (2015): 323–38. http://dx.doi.org/10.1239/jap/1437658601.

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We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δtγ) for any γ < ½. We also prove that the scheme converges uniformly in the strong Lp-sense but with no rate given.

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Kovács, Mihály, Stig Larsson, and Fredrik Lindgren. "On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation." Journal of Applied Probability 52, no. 02 (2015): 323–38. http://dx.doi.org/10.1017/s0021900200012493.

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We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt γ) for any γ &lt; ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.

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37

Jiang, Ziwen, and Deren Xie. "The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar." Abstract and Applied Analysis 2012 (2012): 1–23. http://dx.doi.org/10.1155/2012/475801.

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We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates inL2andH1norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solutionuinH1norm. Numerical experiment illustrates the convergence and stability of this scheme.

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38

Cuesta, E., M. P. Calvo, and C. Palencia. "BACKWARD EULER METHOD AS A POSITIVITY PRESERVING METHOD FOR ABSTRACT INTEGRAL EQUATIONS OF CONVOLUTION TYPE." IFAC Proceedings Volumes 39, no. 11 (2006): 517–21. http://dx.doi.org/10.3182/20060719-3-pt-4902.00086.

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Tang, Tao, Weidong Zhao, and Tao Zhou. "Deferred Correction Methods for Forward Backward Stochastic Differential Equations." Numerical Mathematics: Theory, Methods and Applications 10, no. 2 (2017): 222–42. http://dx.doi.org/10.4208/nmtma.2017.s02.

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AbstractThe deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement

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40

Elizabeth, Sebastian, and Victor Preethi. "A MATHEMATICAL MODEL ON INDUSTRIALIZATION DUE TO HUMAN POPULATION." International Journal of Current Research and Modern Education, Special Issue (August 13, 2017): 38–42. https://doi.org/10.5281/zenodo.842202.

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In this paper, a nonlinear mathematical model is proposed and analyzed to study the Industrialization due to human population. We consider the variables namely, density of the human population and the Industries. We consider that the growth rate of industries is dependent on the density of human population. It is noted that for sustained industrialization, control measures on its growth are required to maintain the ecological stability. We discretize the model by applying Backward Euler method and analyse the stability of the model. Finally, we provide some numerical simulations using MATLAB.

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41

Huang, Zhejun, Huiyun Li, Wenfei Li, et al. "A New Trajectory Tracking Algorithm for Autonomous Vehicles Based on Model Predictive Control." Sensors 21, no. 21 (2021): 7165. http://dx.doi.org/10.3390/s21217165.

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Trajectory tracking is a key technology for precisely controlling autonomous vehicles. In this paper, we propose a trajectory-tracking method based on model predictive control. Instead of using the forward Euler integration method, the backward Euler integration method is used to establish the predictive model. To meet the real-time requirement, a constraint is imposed on the control law and the warm-start technique is employed. The MPC-based controller is proved to be stable. The simulation results demonstrate that, at the cost of no or a little increase in computational time, the tracking pe

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42

Shmaliy, Yuriy S., Shunyi Zhao, and Choon K. Ahn. "Kalman and UFIR state estimation with coloured measurement noise using backward Euler method." IET Signal Processing 14, no. 2 (2020): 64–71. http://dx.doi.org/10.1049/iet-spr.2019.0166.

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43

Liu, Tianyun, Chongbin Zhao, Qingbin Li, and Lihong Zhang. "An efficient backward Euler time-integration method for nonlinear dynamic analysis of structures." Computers & Structures 106-107 (September 2012): 20–28. http://dx.doi.org/10.1016/j.compstruc.2012.03.019.

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44

Timsina, Ramesh Chandra, and Kedar Nath Uprety. "Comparison of Finite Difference Schemes for Fluid Flow in Unsaturated Porous Medium (Soil)." Nepali Mathematical Sciences Report 39, no. 1 (2022): 22–35. http://dx.doi.org/10.3126/nmsr.v39i1.46915.

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Water movement in an unsaturated porous medium (soil) can be expressed by Richards equation with the mass conservation law and Darcy-Buckingham's law. This equation can be expressed in three different forms as pressure head-based, moisture content based and mixed from. In this study, we solve one dimensional Richards Equation in mixed form numerically using finite difference method with various time-stepping schemes: Forward Euler, Backward Euler, Crank-Nicolson and a Stabilized Runge-Kutta-Legendre Super Time-Stepping and we compare their performances using Dirichlet boundary condition on an

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Shen, Wensheng, Changjiang Zhang, Jun Zhang, and Xiaoqian Ma. "Newton Method for Nonlinear Dynamic Systems with Adaptive Time Stepping." JUCS - Journal of Universal Computer Science 16, no. (6) (2010): 891–902. https://doi.org/10.3217/jucs-016-06-0891.

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This paper presents a nonlinear solver based on the Newton-Krylov methods, where the Newton equations are solved by Krylov-subspace type approaches. We focus on the solution of unsteady systems, in which the temporal terms are discretized by the backward Euler method using finite difference. To save computational cost, an adaptive time stepping is used to minimize the number of time steps. The developed program can be applied to solve any nonlinear equations, provided the users could supply the discrete form of the equations. In particular, the nonlinear solver is implemented to solve unsteady

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Jana Aksah, Saufianim, and Zarina Bibi Ibrahim. "Singly Diagonally Implicit Block Backward Differentiation Formulas for HIV Infection of CD4+T Cells." Symmetry 11, no. 5 (2019): 625. http://dx.doi.org/10.3390/sym11050625.

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In this study, a singly diagonally implicit block backward differentiation formula (SDIBBDF) was proposed to approximate solutions for a dynamical HIV infection model of CD 4 + T cells. A SDIBBDF method was developed to overcome difficulty when implementing the fully implicit method by deriving the proposed method in lower triangular form with equal diagonal coefficients. A comparative analysis between the proposed method, BBDF, classical Euler, fourth-order Runge-Kutta (RK4) method, and a Matlab solver was conducted. The numerical results proved that the SDIBBDF method was more efficient in s

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Zeng, Yuping, Kunwen Wen, Fen Liang, and Huijian Zhu. "A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation." Mathematical Problems in Engineering 2020 (April 17, 2020): 1–11. http://dx.doi.org/10.1155/2020/7525676.

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We introduce and analyze a weakly overpenalized symmetric interior penalty method for solving the heat equation. We first provide optimal a priori error estimates in the energy norm for the fully discrete scheme with backward Euler time-stepping. In addition, we apply elliptic reconstruction techniques to derive a posteriori error estimators, which can be used to design adaptive algorithms. Finally, we present two numerical experiments to validate our theoretical analysis.

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Yu, Zhanhua. "Almost Surely Asymptotic Stability of Exact and Numerical Solutions for Neutral Stochastic Pantograph Equations." Abstract and Applied Analysis 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/143079.

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We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs), and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM) with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.

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Dautov, Rafail Z., and Alexander V. Lapin. "Sharp error estimate for implicit finite element scheme for American put option." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 2 (2019): 85–94. http://dx.doi.org/10.1515/rnam-2019-0007.

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AbstractWe investigate a numerical solution method for a degenerate parabolic variational inequality that determines American vanilla put pricing. This method is based on piecewise linear finite elements in spatial variables and the backward Euler finite difference in time variable. For the approximate solution, we get sharp error estimate of orderO(h+τ3/4) in the energy norm of the corresponding differential operator.

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Pani, Amiya K., Vidar Thomée, and A. S. Vasudeva Murthy. "A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem." Computational Methods in Applied Mathematics 20, no. 4 (2020): 769–82. http://dx.doi.org/10.1515/cmam-2020-0009.

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AbstractWe 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\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bou

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