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Journal articles on the topic 'Implicit Euler Method'

Author: Grafiati

Published: 3 June 2025

Last updated: 23 June 2025

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1

Zhang, Gui-Lai, Zhi-Yong Zhu, Lei-Ke Chen, and Song-Shu Liu. "Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Pulse Vaccination Strategy." Axioms 13, no. 12 (2024): 854. https://doi.org/10.3390/axioms13120854.

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In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions.

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2

Herdiana, Ratna. "NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS USING IMPLICIT MILSTEIN METHOD." Journal of Fundamental Mathematics and Applications (JFMA) 3, no. 1 (2020): 72–83. http://dx.doi.org/10.14710/jfma.v3i1.7416.

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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.

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3

Dlamini, P. G., and M. Khumalo. "On the Computation of Blow-Up Solutions for Nonlinear Volterra Integrodifferential Equations." Mathematical Problems in Engineering 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/878497.

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We make use of an adaptive numerical method to compute blow-up solutions for nonlinear ordinary Volterra integrodifferential equations (VIDEs). The method is based on the implicit midpoint method and the implicit Euler method and is named the implicit midpoint-implicit Euler (IMIE) method and was used to compute blow-up solutions in semilinear ODEs and parabolic PDEs in our earlier work. We demonstrate that the method produces superior results to the adaptive PECE-implicit Euler (PECE-IE) method and the MATLAB solver of comparable order just as it did in our previous contribution. We use quadrature rules to approximate the integral in the VIDE and demonstrate that the choice of quadrature rule has a significant effect on the blow-up time computed. In cases where the problem contains a convolution kernel with a singularity we use convolution quadrature.

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4

Xu, Zhi-Wei, and Gui-Lai Zhang. "Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination." Axioms 14, no. 6 (2025): 470. https://doi.org/10.3390/axioms14060470.

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This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results.

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5

Thomée, Vidar, and A. S. Vasudeva Murthy. "An Explicit-Implicit Splitting Method for a Convection-Diffusion Problem." Computational Methods in Applied Mathematics 19, no. 2 (2019): 283–93. http://dx.doi.org/10.1515/cmam-2018-0018.

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AbstractWe analyze a second-order accurate finite difference method for a spatially periodic convection-diffusion problem. The method is a time stepping method based on the Strang splitting of the spatially semidiscrete solution, in which the diffusion part uses the Crank–Nicolson method and the convection part the explicit forward Euler approximation on a shorter time interval. When the diffusion coefficient is small, the forward Euler method may be used also for the diffusion term.

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6

Dlamini, P. G., and M. Khumalo. "On the Computation of Blow-up Solutions for Semilinear ODEs and Parabolic PDEs." Mathematical Problems in Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/162034.

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We introduce an adaptive numerical method for computing blow-up solutions for ODEs and well-known reaction-diffusion equations. The method is based on the implicit midpoint method and the implicit Euler method. We demonstrate that the method produces superior results to the adaptive PECE-implicit method and the MATLAB solver of comparable order.

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7

Zhao, Shibo. "The advantage of symplectic Euler in optimization and its application." Theoretical and Natural Science 10, no. 1 (2023): 230–34. http://dx.doi.org/10.54254/2753-8818/10/20230349.

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Many professions today place a high value on optimization, and many problems can eventually be transformed into optimization issues. There are many iterative methods available today to handle optimization issues, however many algorithms' design principles are unclear. Weijie Su solved this problem by discretizing the iterative equation using an ordinary differential equation, but different discretization techniques will provide different outcomes. So choosing an appropriate method is important. Three discretization techniquesexplicit Euler, implicit Euler, and symplectic Eulerare compared in this work. It is found that while both symplectic and implicit Euler can accelerate the process, only symplectic Euler can be put to use in practice. This further demonstrates symplectic Euler's supremacy in iteration. The use of symplectic Euler in other fields is also introduced in this study, particularly in the Lotka-Volterra equation where promising results might be attained. Symplectic Euler is critical to optimization and is likely to be applied in more areas in the future.

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8

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 infectious illnesses. The SIR Model was easily solved using the Adams Bashforth 2-step analysis and the Backward Euler method. The approaches' solutions are close to the exact requirements. There are important distinctions between the two-step Adams Bashforth and backward Euler procedures. The running time of the Adams Bashforth 2-step backward Euler method is shorter than that of the backward Euler method.

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9

Karpaev, Alexey Alexeevich, and Rubin Renatovich Aliev. "Application of simplified implicit Euler method for electrophysiological models." Computer Research and Modeling 12, no. 4 (2020): 845–64. http://dx.doi.org/10.20537/2076-7633-2020-12-4-845-864.

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10

Czernous, Wojciech. "Generalized implicit Euler method for hyperbolic functional differential equations." Mathematische Nachrichten 283, no. 8 (2010): 1114–33. http://dx.doi.org/10.1002/mana.200710067.

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11

MUNTEANU, MARILENA, and LUCA F. PAVARINO. "DECOUPLED SCHWARZ ALGORITHMS FOR IMPLICIT DISCRETIZATIONS OF NONLINEAR MONODOMAIN AND BIDOMAIN SYSTEMS." Mathematical Models and Methods in Applied Sciences 19, no. 07 (2009): 1065–97. http://dx.doi.org/10.1142/s0218202509003723.

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Novel decoupled Schwarz algorithms for the implicit discretizations of the Monodomain and Bidomain systems in three dimensions are constructed and analyzed. Both implicit Euler and linearly implicit Rosenbrock time discretizations are considered. Convergence rate estimates are proven for a domain decomposition preconditioner based on overlapping additive Schwarz techniques and employed in a Newton–Krylov–Schwarz method for the Euler scheme. An analogous result is proven for the same preconditioner applied to the linear systems originated in the Rosenbrock scheme. Several parallel numerical results in three dimensions confirm the convergence rates predicted by the theory and study the performance of our algorithms for a complete heartbeat, for both Bidomain and Monodomain models, Euler and Rosenbrock schemes, fixed time step and adaptive strategies. The results also show the considerable CPU-time reduction of our decoupled Schwarz solvers with respect to fully-implicit Schwarz solvers.

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12

Hasan, M. Kamrul, M. Suzan Ahamed, M. S. Alam, and M. Bellal Hossain. "An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems." Journal of Computational Engineering 2013 (September 26, 2013): 1–5. http://dx.doi.org/10.1155/2013/720812.

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An implicit method has been presented for solving singular initial value problems. The method is simple and gives more accurate solution than the implicit Euler method as well as the second order implicit Runge-Kutta (RK2) (i.e., implicit midpoint rule) method for some particular singular problems. Diagonally implicit Runge-Kutta (DIRK) method is suitable for solving stiff problems. But, the derivation as well as utilization of this method is laborious. Sometimes it gives almost similar solution to the two-stage third order diagonally implicit Runge-Kutta (DIRK3) method and the five-stage fifth order diagonally implicit Runge-Kutta (DIRK5) method. The advantage of the present method is that it is used with less computational effort.

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13

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 proposed methods, numerical examples are considered for various linear and nonlinear fractional differential equations with different kernels. The results show that theses numerical schemes are feasible in application.</p></abstract>

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14

Nabizadeh, Mohammad Sina, Ritoban Roy-Chowdhury, Hang Yin, Ravi Ramamoorthi, and Albert Chern. "Fluid Implicit Particles on Coadjoint Orbits." ACM Transactions on Graphics 43, no. 6 (2024): 1–38. http://dx.doi.org/10.1145/3687970.

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We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.

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15

Alomari, Mohammad W., Iqbal M. Batiha, Wala’a Ahmad Alkasasbeh, Nidal Anakira, Iqbal H. Jebril, and Shaher Momani. "Euler-Maclaurin Method for Approximating Solutions of Initial Value Problems." International Journal of Robotics and Control Systems 5, no. 1 (2025): 366–80. https://doi.org/10.31763/ijrcs.v5i1.1560.

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This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by the Euler-Maclaurin formula. This results in a higher-order implicit corrected method that outperforms Taylor’s and Runge–Katta’s methods in terms of accuracy. We derive an error bound for the Euler-Maclaurin higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Taylor and Runge-Katta methods. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficiency of our proposed method over the traditional well-known methods.

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16

Trompert, R. A., and J. G. Verwer. "Analysis of the Implicit Euler Local Uniform Grid Refinement Method." SIAM Journal on Scientific Computing 14, no. 2 (1993): 259–78. http://dx.doi.org/10.1137/0914017.

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17

Ouaar, Fatima. "Neural network and numerical methods performance comparison for prey-predator model." South Florida Journal of Development 5, no. 7 (2024): e4211. http://dx.doi.org/10.46932/sfjdv5n7-041.

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In several areas, differential equations are used extensively to simulate a wide range of events. The Prey-Predator model, sometimes referred to the Lotka-Volterra equations, was used as an example in this work. On the other hand, occasionally insufficient data is available to build an explicit model for this problem. Therefore, being able to approximate differential equation solutions is important. This paper's primary contribution is the performance comparison between the implicit Euler approach and the neural network method. The outcomes demonstrate that although the neural network approach takes longer to provide an estimate, it consistently produces better estimates than the implicit Euler technique.

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18

Quyet, Dao Trong. "Spectral Galerkin Method in Space and Time for the 2Dg-Navier-Stokes Equations." Abstract and Applied Analysis 2013 (2013): 1–18. http://dx.doi.org/10.1155/2013/805685.

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We prove theH2-stability andL2-error analysis of the spectral Galerkin method in space and time with the implicit/explicit Euler scheme for the 2Dg-Navier-Stokes equations in bounded domains when the initial data belong toH1.

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19

Iqbal, Iqbal, Mohammad W. Alomari, Nidal Anakira, Saad Meqdad, Iqbal H. Jebril, and Shaher Momani. "Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem." International Journal of Neutrosophic Science 25, no. 3 (2025): 76–91. http://dx.doi.org/10.54216/ijns.250308.

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This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by the Euler-Maclaurin formula. This results in a higher-order implicit corrected method that outperforms the Runge-Kutta method in terms of accuracy. We derive an error bound for the Euler-Maclaurin higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Runge-Kutta method. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficiency of our proposed method over the traditional well-known methods. In conclusion, the proposed method consistently outperforms the Runge-Kutta method experimentally in all practical problems.

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20

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 bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {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.

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21

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 solving the model than the methods to be compared.

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22

Brenneis, A., and A. Eberle. "Application of an Implicit Relaxation Method Solving the Euler Equations for Time-Accurate Unsteady Problems." Journal of Fluids Engineering 112, no. 4 (1990): 510–20. http://dx.doi.org/10.1115/1.2909436.

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A numerical procedure is presented for computing time-accurate solutions of flows about two and three-dimensional configurations using the Euler equations in conservative form. A nonlinear Newton method is applied to solve the unfactored implicit equations. Relaxation is performed with a point Gauss-Seidel algorithm ensuring a high degree of vectorization by employing the so-called checkerboard scheme. The fundamental feature of the Euler solver is a characteristic variable splitting scheme (Godunov-type averaging procedure, linear locally one-dimensional Riemann solver) based on an eigenvalue analysis for the calculation of the fluxes. The true Jacobians of the fluxes on the right-hand side are used on the left-hand side of the first order in time-discretized Euler equations. A simple matrix conditioning needing only few operations is employed to evade singular behavior of the coefficient matrix. Numerical results are presented for transonic flows about harmonically pitching airfoils and wings. Comparisons with experiments show good agreement except in regions where viscous effects are evident.

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23

Xiong, Xiaogang, Wei Chen, Shanhai Jin, and Shyam Kamal. "Discrete-Time Implementation of Continuous Terminal Algorithm With Implicit-Euler Method." IEEE Access 7 (2019): 175940–46. http://dx.doi.org/10.1109/access.2019.2957282.

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24

Santos, Rômulo Damasclin Chaves dos, Jorge Henrique de Oliveira Sales, and Alice Rosa da Silva. "A mathematical analysis to the approximate weak solution of the Smagorinsky Model for different flow regimes." Journal of Engineering and Exact Sciences 10, no. 1 (2024): 17579. http://dx.doi.org/10.18540/jcecvl10iss01pp17579.

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This study delves into the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes, employing the Smagorinsky Model (SM). By treating the model as inherently discrete, we implement a semi-implicit time discretization using the Euler method. This approach includes comprehensive stability analyses, applicable to a spectrum of flow regimes, and an exploration of the asymptotic energy balance dynamics during fluid movements. The primary contribution of this study is found in its methodical approach to the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes using the Smagorinsky Model (SM). The adoption of a semi-implicit time discretization with the Euler method, coupled with a meticulous analysis of energy balance, establishes a robust foundation adaptable to diverse flow conditions.

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25

Zhong Shuang-Ying and Wu Xin. "Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule." Acta Physica Sinica 60, no. 9 (2011): 090402. http://dx.doi.org/10.7498/aps.60.090402.

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26

Chai, Shimin, and Yongkui Zou. "The Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility." Journal of Applied Mathematics 2012 (2012): 1–35. http://dx.doi.org/10.1155/2012/808216.

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This paper is concerned with the numerical approximations of the Cahn-Hilliard-type equation with concentration-dependent mobility. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method for the space and the implicit Euler method for the time. Numerical experiments are carried out to illustrate the theoretical analysis.

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27

ZHANG, RONGXIN, GUOLIANG QIN, and CHANGYUN ZHU. "SPECTRAL ELEMENT METHOD FOR ACOUSTIC PROPAGATION PROBLEMS BASED ON LINEARIZED EULER EQUATIONS." Journal of Computational Acoustics 17, no. 04 (2009): 383–402. http://dx.doi.org/10.1142/s0218396x09004014.

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A Chebyshev spectral element approximation of acoustic propagation problems based on linearized Euler equations is introduced, and the numerical approach is based on spectral elements in space with first-order Clayton–Engquist–Majda absorbing boundary conditions and implicit Newmark method in time. An initial perturbation problem has been solved to test the accuracy and stability of the numerical method. Then the sound propagation by source terms is also studied, including the radiation of a monopole and dipolar source in both stationary medium and uniform mean flow. The numerical simulation leads to good results in both accuracy and stability. Compared with the analytical solutions, the numerical results show the advantages in spectral accuracy even with relatively fewer grid points. Moreover, the implicit Newmark method in time marching also presents its superiority in stability. Finally, a problem of sound propagation in pipes is simulated as well.

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28

SI, HAI-QING, TONG-GUANG WANG, and XIAO-YUN LUO. "A FULLY IMPLICIT SOLVER OF 3-D EULER EQUATIONS ON MULTIBLOCK CURVILINEAR GRIDS." Modern Physics Letters B 19, no. 28n29 (2005): 1483–86. http://dx.doi.org/10.1142/s0217984905009717.

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A fully implicit unfactored algorithm for three-dimensional Euler equations is developed and tested on multi-block curvilinear meshes. The convective terms are discretized using an upwind TVD scheme. The large sparse linear system generated at each implicit time step is solved by GMRES* method combined with the block incomplete lower-upper preconditioner. In order to reduce the memory requirements and the matrix-vector operation counts, an approximate method is used to derive the Jacobian matrix, which only costs half of the computational efforts of the exact Jacobian calculation. The comparison between the numerical results and the experimental data shows good agreement, which demonstrates that the implicit algorithm presented is effective and efficient.

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29

Al Ghafli, Ahmed A., Yasir Nawaz, Hassan J. Al Salman, and Muavia Mansoor. "Extended Runge-Kutta Scheme and Neural Network Approach for SEIR Epidemic Model with Convex Incidence Rate." Processes 11, no. 9 (2023): 2518. http://dx.doi.org/10.3390/pr11092518.

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For solving first-order linear and nonlinear differential equations, a new two-stage implicit–explicit approach is given. The scheme’s first stage, or predictor stage, is implicit, while the scheme’s second stage is explicit. The first stage of the proposed scheme is an extended form of the existing Runge–Kutta scheme. The scheme’s stability and consistency are also offered. In two phases, the technique achieves third-order accuracy. The method is applied to the SEIR epidemic model with a convex incidence rate. The local stability is also examined. The technique is evaluated compared to existing Euler and nonstandard finite difference methods. In terms of accuracy, the produced plots show that the suggested scheme outperforms the existing Euler and nonstandard finite difference methods. Furthermore, a neural network technique is being considered to map the relationship between time and the amount of susceptible, exposed, and infected people.

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Yu, Yuexin, and Sheng Zhen. "Convergence Analysis of Implicit Euler Method for a Class of Nonlinear Impulsive Fractional Differential Equations." Mathematical Problems in Engineering 2020 (December 8, 2020): 1–8. http://dx.doi.org/10.1155/2020/8826338.

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For a class of nonlinear impulsive fractional differential equations, we first transform them into equivalent integral equations, and then the implicit Euler method is adapted for solving the problem. The convergence analysis of the method shows that the method is convergent of the first order. The numerical results verify the correctness of the theoretical results.

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CHINVIRIYASIT, W., and S. CHINVIRIYASIT. "NUMERICAL SOLUTION OF NONLINEAR EQUATION TO COMBINED DETERMINISTIC AND NARROW-BAND RANDOM EXCITATION." International Journal of Modern Physics B 22, no. 21 (2008): 3655–75. http://dx.doi.org/10.1142/s0217979208048528.

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The Duffing oscillator to combined deterministic and narrow-band random excitation, which is a nonlinear equation, is studied and solved numerically using three numerical methods based on finite difference schemes. Method 1, the well-known Euler method, is an explicit method; Method 2 is an implicit first-order method which does not bring contrived chaos into the solution; and Method 3 is based on two first-order methods which is second-order method and is chaos-free. In a series of numerical experiments, it is seen that the proposed methods have superior stability properties to those of the well-known Euler and fourth-order Runge-Kutta methods to which they are compared. When extended to the numerical solution of Duffing oscillator to combined deterministic and narrow-band random excitation, the developed methods give the correct steady-state solutions compared with the literature.

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Aleksandrov, Alexander Yu. "Application of the implicit Euler method for the discretization of some classes of nonlinear systems." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 19, no. 3 (2023): 304–19. http://dx.doi.org/10.21638/11701/spbu10.2023.301.

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The problem of stability preservation under discretization of some classes of nonlinear differential equations systems is studied. Persidskii systems, Lurie systems of indirect control, and systems whose right-hand sides have a canonical structure are considered. It is assumed that the zero solutions of these systems are globally asymptotically stable. Conditions are determined that guarantee the asymptotic stability of the zero solutions for the corresponding difference systems. Previously, such conditions were established for the case where discretization was carried out using the explicit Euler method. In this paper, difference schemes are constructed on the basis of the implicit Euler method. For the obtained discrete systems, theorems on local and global asymptotic stability are proved, estimates of the time of transient processes are derived. For systems with a canonical structure of right-hand sides, based on the approach of V. I. Zubov, a modified implicit computational scheme is proposed that ensures the matching of the convergence rate of solutions to the origin for the differential and corresponding difference systems. It is shown that implicit computational schemes can guarantee the preservation of asymptotic stability under less stringent constraints on the discretization step and right-hand sides of the systems under consideration compared to the constraints obtained using the explicit method. An example is presented illustrating the obtained theoretical conclusions.

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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 formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

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Bonfiglioli, A., P. De Palma, G. Pascazio, and M. Napolitano. "An Implicit Fluctuation Splitting Scheme for Turbomachinery Flows." Journal of Turbomachinery 127, no. 2 (2005): 395–401. http://dx.doi.org/10.1115/1.1777576.

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This paper describes an accurate, robust and efficient methodology for solving two-dimensional steady transonic turbomachinery flows. The Euler fluxes are discretized in space using a hybrid multidimensional upwind method, which, according to the local flow conditions, uses the most suitable fluctuation splitting (FS) scheme at each cell of the computational domain. The viscous terms are discretized using a standard Galerkin finite element scheme. The eddy viscosity is evaluated by means of the Spalart-Allmaras turbulence transport equation, which is discretized in space by means of a mixed FS-Galerkin approach. The equations are discretized in time using an implicit Euler scheme, the Jacobian being evaluated by two-point backward differences. The resulting large sparse linear systems are solved sequentially using a preconditioned GMRES strategy. The proposed methodology is employed to compute subsonic and transonic turbulent flows inside a high-turning turbine-rotor cascade.

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Dubuc, L., F. Cantariti, M. Woodgate, B. Gribben, K. J. Badcock, and B. E. Richards. "Solution of the Unsteady Euler Equations Using an Implicit Dual-Time Method." AIAA Journal 36, no. 8 (1998): 1417–24. http://dx.doi.org/10.2514/2.532.

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Spigler, Renato, and Marco Vianello. "Convergence analysis of the semi-implicit euler method for abstract evolution equations." Numerical Functional Analysis and Optimization 16, no. 5-6 (1995): 785–803. http://dx.doi.org/10.1080/01630569508816645.

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Wang, La-sheng, Changlin Mei, and Hong Xue. "The semi-implicit Euler method for stochastic differential delay equation with jumps." Applied Mathematics and Computation 192, no. 2 (2007): 567–78. http://dx.doi.org/10.1016/j.amc.2007.03.027.

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Dubuc, L., F. Cantariti, M. Woodgate, B. Gribben, K. J. Badcock, and B. E. Richards. "Solution of the unsteady Euler equations using an implicit dual-time method." AIAA Journal 36 (January 1998): 1417–24. http://dx.doi.org/10.2514/3.13984.

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Li, Limei, and Da Xu. "Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation." Journal of Computational Physics 236 (March 2013): 157–68. http://dx.doi.org/10.1016/j.jcp.2012.11.005.

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Luo, Dong, Xiaogang Xiong, Shanhai Jin, and Wei Chen. "Implicit Euler Implementation of Twisting Controller and Super-Twisting Observer without Numerical Chattering: Precise Quasi-Static MEMS Mirrors Control." MATEC Web of Conferences 256 (2019): 03004. http://dx.doi.org/10.1051/matecconf/201925603004.

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The quasi-static operations of MEMS mirror are very sensitive to undesired oscillations due to its very low damping. It has been shown that closed-loop control can be superior to reduce those oscillations than open-loop control in the literature. For the closed-loop control, the conventional way of implementing sliding mode control (SMC) algorithm is forward Euler method, which results in numerical chattering in the control input and output. This paper proposes an implicit Euler implementation scheme of super twisting observer and twisting control for a commercial MEMS mirror actuated by an electrostatic staggered vertical comb (SVC) drive structure. The famous super-twisting algorithm is used as an observer and twisting SMC is used as a controller. Both are discretized by an implicit Euler integration method, and their implementation algorithms are provided. Simulations verify that, as compared to traditional sliding mode control implementation, the proposed scheme reduces the chattering both in trajectory tracking output and control input in presence of model uncertainties and external disturbances. The comparison demonstrates the potential applications of the proposed scheme in industrial applications in terms of feasibility and performance.

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Kadalbajoo, Mohan K., and Ashish Awasthi. "Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh." Asian-European Journal of Mathematics 10, no. 02 (2016): 1750029. http://dx.doi.org/10.1142/s1793557117500292.

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In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.

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Glazyrina, O. V., R. Z. Dautov, and D. A. Gubaidullina. "Accuracy of an implicit scheme for the finite element method with a penalty for a nonlocal parabolic obstacle problem." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 2 (March 11, 2024): 3–21. http://dx.doi.org/10.26907/0021-3446-2024-2-3-21.

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In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.

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Guliyeva, C. "COMPARISON OF EXPLICIT RUNGE-KUTTA METHODS WITH HYBRID METHODS OF MULTISTEP TYPE." Scientific heritage, no. 155 (February 23, 2025): 66–69. https://doi.org/10.5281/zenodo.14914574.

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As is known, in solving some of applied problems, Runge-Kutta methods are used, considering that these methods are explicit and do not require the use of any other methods. Note that, like other methods, the Runge-Kutta methods also have their own advantages and disadvantages. One of the main disadvantages of Runge-Kutta method consists of repeatedly calculating the function , which is right-hand side of the differential equation, and one of the advantages of Runge-Kutta methods is that they are one step methods. Usually, such methods are considered as the recurrent relationships. A well-known representative of the Runge-Kutta method is the Euler and Midpoint methods. It is known that one of the exact methods is called the hybrid method, which can be taken as the generalization of the Midpoint method. Consequently, the Runge-Kutta and Hybrid methods are generalizations of this method. By using this, here we consider the comparison of some specific Runge-Kutta methods with the corresponding hybrid methods. Since implicit hybrid methods are more exact, it becomes necessary to compare hybrid methods with the semi-implicit Runge-Kutta methods. By considering a specific method, we compare semi-implicit Runge-Kutta methods with implicit hybrid methods.

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Gheorghiu, Călin-Ioan. "On some one-step implicit methods as dynamical systems." Journal of Numerical Analysis and Approximation Theory 32, no. 2 (2003): 171–75. http://dx.doi.org/10.33993/jnaat322-745.

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The one-step implicit methods, the backward Euler being the most known, require the solution of a nonlinear equation at each step.To avoid this, these methods can be approximated by making use of a one step of a Newton method. Thus the methods are transformed into some explicit ones. We will obtain these transformed methods, find conditions under which they generate continuous dynamical systems and show their order of convergence. Some results on the stability of these explicit schemes, as well as on the shadowing phenomenon are also carried out. Concluding remarks and some open problems end the paper.

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Alomari, Mohammad W., Iqbal M. Batiha, Abeer Al-Nana, Mohammad Odeh, Nidal Anakira, and Shaher Momani. "A Comparative Analysis of Numerical Techniques: Euler-Maclaurin vs. Runge-Kutta Methods." Journal of Robotics and Control (JRC) 6, no. 2 (2025): 812–21. https://doi.org/10.18196/jrc.v6i2.25566.

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This study introduces a novel higher-order implicit correction method derived from the Euler-Maclaurin formula to enhance the approximation of initial value problems. The proposed method surpasses the Runge-Kutta approach in accuracy, stability, and convergence. An error bound is established to demonstrate its theoretical reliability. To validate its effectiveness, numerical experiments are conducted, showcasing its superior performance compared to conventional methods. The results consistently confirm that the proposed method outperforms the Runge-Kutta method across various practical applications.

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Goura, G. S. L., K. J. Badcock, M. A. Woodgate, and B. E. Richards. "Implicit method for the time marching analysis of flutter." Aeronautical Journal 105, no. 1046 (2001): 199–214. http://dx.doi.org/10.1017/s0001924000025446.

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Abstract This paper evaluates a time marching simulation method for flutter which is based on a solution of the Euler equations and a linear modal structural model. Jameson’s pseudo time method is used for the time stepping, allowing sequencing errors to be avoided without incurring additional computational cost. Transfinite interpolation of displacements is used for grid regeneration and a constant volume transformation for inter-grid interpolation. The flow pseudo steady state is calculated using an unfactored implicit method which features a Krylov subspace solution of an approximately linearised system. The spatial discretisation is made using Osher’s approximate Riemann solver with MUSCL interpolation. The method is evaluated against available results for the AGARD 445.6 wing. This wing, which is made of laminated mahogany, was tested at NASA Langley in the 1960s and has been the standard test case for simulation methods ever since. The structural model in the current work was built in NASTRAN using homogeneous plate elements. The comparisons show good agreement for the prediction of flutter boundaries. The solution method allows larger time steps to be taken than other methods.

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Daba, Imiru Takele, Wondwosen Gebeyaw Melesse, and Guta Demisu Kebede. "A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay." Computational and Mathematical Methods 2023 (October 25, 2023): 1–12. http://dx.doi.org/10.1155/2023/6496354.

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This paper is aimed at constructing and analyzing a fitted approach for singularly perturbed time delay parabolic problems with two small parameters. The proposed computational scheme comprises the implicit Euler and especially finite difference method for the time and space variable discretization, respectively, on uniform step size. The stability and convergence analysis of the method is provided and is first-order parameter uniform convergent. Further, the numerical results depict that the present method is more convergent than some methods available in the literature.

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BOFFI, DANIELE, LUCIA GASTALDI, and LUCA HELTAI. "NUMERICAL STABILITY OF THE FINITE ELEMENT IMMERSED BOUNDARY METHOD." Mathematical Models and Methods in Applied Sciences 17, no. 10 (2007): 1479–505. http://dx.doi.org/10.1142/s0218202507002352.

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The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions. Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method. A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly. We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.

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Yang, Zhanwen, Tianqing Zuo, and Zhijie Chen. "Numerical analysis of linearly implicit Euler-Riemann method for nonlinear Gurtin-MacCamy model." Applied Numerical Mathematics 163 (May 2021): 147–66. http://dx.doi.org/10.1016/j.apnum.2020.12.018.

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CHEN, H. Q., and C. SHU. "AN EFFICIENT IMPLICIT MESH-FREE METHOD TO SOLVE TWO-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS." International Journal of Modern Physics C 16, no. 03 (2005): 439–54. http://dx.doi.org/10.1142/s0129183105007327.

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Local radial basis function-based differential quadrature (RBF-DQ) method is a natural mesh-free approach, in which any derivative of a function at a point is approximated by a weighted linear sum of functional values at its surrounding scattered points. In this paper, the weighting coefficients in the spatial derivative approximation of the Euler equation are determined by using a weighted least-square procedure in the frame of RBFs, which enhances the flexibility of distributing points in the computational domain. An upwind method is further introduced to cope with discontinuities by using Roe's approximate Riemann solver for estimation of the inviscid flux on the virtual mid-point between the reference knot and its surrounding knot. The lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm, which is implemented in a matrix-free form like the one used in the finite-volume method, is introduced in the work to speed up the convergence. The proposed approach is validated by its application to simulate transonic flows over a NACA 0012 airfoil. It was found that the present mesh-free results agree very well with available data in the literature, and the implicit LU-SGS algorithm can greatly save the computational time as compared with explicit time marching methods.

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