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Academic literature on the topic 'Implicit Euler Method'

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Published: 3 June 2025

Last updated: 23 June 2025

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Journal articles on the topic "Implicit Euler Method"

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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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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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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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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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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Dissertations / Theses on the topic "Implicit Euler Method"

1

Barnes, Caleb J. "An Implicit High-Order Spectral Difference Method for the Compressible Navier-Stokes Equations Using Adaptive Polynomial Refinement." Wright State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=wright1315591802.

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Jebens, Stefan Verfasser], Rüdiger [Akademischer Betreuer] [Weiner, and Jens [Akademischer Betreuer] Lang. "Explicit and linearly implicit peer methods for the solution of the compressible Euler equations / Stefan Jebens. Betreuer: Rüdiger Weiner ; Jens Lang." Halle, Saale : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2011. http://d-nb.info/1025231295/34.

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Stissi, Santina Chiara. "Ghost-point methods for Elliptic and Hyperbolic Equations." Doctoral thesis, Università di Catania, 2019. http://hdl.handle.net/10761/4104.

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In this thesis we have presented a finite-difference ghost-point method to solve elliptic and hyperbolic equations on arbitrary domains. The equations are discretized on a uniform Cartesian grid. At first we applied the Coco-Russo method, which represents a generalization of the finite-difference method for the elliptic equations on arbitrary domains, at the resolution of the Poisson equation. This method proposes a polynomial interpolation technique to impose boundary conditions and therefore the interpolation error can influence the accuracy order of the method itself. We have proposed linear and bilinear interpolation techniques. These conditions are imposed on the projections of the ghost points on the border of the domain. The numerical tests performed on the behaviors of the inverse matrix of the method, of the error and of the consistency error confirm the stability and convergence of the Coco-Russo method in 1D, 2D and 3D, in the case of Dirichlet problems and in the case of mixed problems. We have also presented a rigorous proof of the stability and convergence of the numerical method in the one-dimensional case. Once we certain of the convergence and stability of the Coco-Russo method, our interest it has moved to the study of the Euler equations of the gas dynamic. The Coco-Russo method was applied for the development of a semi-implicit method for Euler equations on domains that have obstacles, to impose boundary conditions in a manner similar to elliptic equations. This method being semi-implicit overcomes the problem of spatial restriction to guarantee the stability of the method typical of explicit methods.

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Diniz, dos Santos Nuno Miguel. "Numerical methods for fluid-structure interaction problems with valves." Paris 6, 2007. http://www.theses.fr/2007PA066683.

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Cette thèse est motivée par la modélisation et la simulation numérique des phénomènes d’interaction fluide-structure autour de valves cardiaques. L’interaction avec la paroi des vaisseaux est traitée avec une formulation Arbitraire Lagrange Euler (ALE), tandis que l’interaction avec les valves est traitée à l’aide de multiplicateurs de Lagrange, dans une formulation de type Domaines Fictifs (FD). Après une présentation de synthèse des di- verses méthodes utilisées en interaction fluide-structure dans les écoulements sanguins, nous décrivons une méthode permettant de simuler la dynamique d’une valve immergée dans un écoulement visqueux incompressible. L’algori- thme de couplage est partionné, ce qui permet de conserver des solveurs fluides et structures indépendants. Le maillage du fluide est mobile pour suivre la paroi des vaisseaux, mais indépendant du maillage des valves. Ceci autorise des très grands déplacements sans nécessiter de remaillage. Nous proposons une stratégie pour gérer le contact entre plusieurs valves. L’algorithme est totalement indépendant des solveurs de structures et est bien adapté au couplage fluide-structure partionné. Enfin, nous proposons un schéma de couplage semi-implicite permettant de méler efficacement les formulations ALE et FD. Toutes les méthodes considérées sont accom- pagnées de nombreux tests numériques en 2D et 3D<br>This thesis is motivated by the modelling and the simulation of fluid-structure interaction phenomena in the vicinity of heart valves. On the one hand, the interaction of the vessel wall is dealt with an Arbitrary Lagrangian Eule- rian (ALE) formulation. On the other hand the interaction of the valves is treated with the help of Lagrange multipliers in a Fictitious Domains-like (FD) formulation. After a synthetic presentation of the several methods available for the fluid-structure interaction in blood flows, we describe a method that permits capture the dynamics of a valve immersed in an in- compressible fluid. The coupling algorithm is partitioned which allows the fluid and structure solvers to remain independent. In order to follow the ves- sel walls, the fluid mesh is mobile, but it remains none the less independent of the valve mesh. In this way we allow large displacements without the need to perform remeshing. We propose a strategy to manage contact between several immersed structures. The algorithm is completely independent of the structure solver and is well adapted to the partitioned fluid-structure coupling. Lastly we propose a semi-implicit coupling scheme allowing to mix, effectively, the ALE and FD formulations. The methods considered are followed with several numerical tests in 2D and 3D

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Scandurra, Leonardo. "Numerical Methods for All Mach Number flows for Gas Dynamics." Doctoral thesis, Università di Catania, 2017. http://hdl.handle.net/10761/4042.

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An original numerical method to solve the all-Mach number flow for the Euler equations of gas dynamics on staggered grid is presented in this thesis. The system is discretized to second order in space on staggered grid, in a fashion similar to the Nessyahu-Tadmor central scheme for 1D model and Jang-Tadmor central scheme for 2D model, thus simplifying the flux computation. This approach turns out to be extremely simple, since it requires no equation splitting. We consider the isentropic case and the general case. For simplicity we assume a gamma-law gas in both cases. Both approaches are based on IMEX strategy, in which some term is treated explicitly, while other terms are treated implicitly, thus avoiding the classical CFL restriction due to acoustic waves. - In Isentropic Euler Case: The core if the implicit term contains a non-linear elliptic equation for the pressure, which has to be treated by a fully implicit technique. Because of the non-linearity, it is necessary to adopt an iterative method to compute the pressure. In our numerical experiments Newton's method worked with few iterations. - General Euler Case: In this case the implicit term is treated in a semi-implicit fashion, thus avoiding the use of Newton's iterations. In both cases the schemes are implemented to second order accuracy in time. Suitably well-prepared initial conditions are considered, which depend on the Mach number. In one space dimension we obtain the same profiles found in the literature for the isentropic case and for the general Euler system for all Mach numbers. In particular, the schemes have been shown to be AP, in the sense that they become a consistent discretizzation of the incompressible Euler equation as the Mach number approaches zero. Numerical evidence of such AP property is provided on a two dimensional test case. The last chapter deals with the piston problem in Lagrangian coordinates treated by a semi-implicit scheme. The implicit treatment of the boundary conditions is originally developed in the thesis. It is shown that for very low Mach number the scheme is able to recover the adiabatic solution with very large CFL numbers. For moderate Mach numbers, or in presence of an initial acoustic wave, loss of accuracy is observerd if the CFL is too large. This drawback can be cured by using a suitable time step control, which will be subject of future investigation. Current work is also related on the development of higher order accurate schemes for 1D and 2D problems.

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Colas, Clément. "Formulation intégrale implicite pour la modélisation d'écoulements fluides en milieu encombré." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0555.

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Le sujet de cette thèse est la modélisation d’écoulements fluides dans des milieux encombrés d’obstacles solides. L’objectif est de proposer une approche intégrale pour la simulation numérique des écoulements dans les composants des circuits thermohydrauliques d’un réacteur nucléaire. Cette approche englobe les échelles de représentation locale et « composant », et assure, par construction, la continuité entre ces deux échelles. Elle repose sur une formulation intégrale multidimensionnelle des équations de la mécanique des fluides qui dégénère naturellement vers l'approche fluide standard de la mécanique des fluides . Sa discrétisation est réalisée avec un schéma numérique aux volumes finis collocalisé en espace et semi-implicite en temps utilisant une méthode de correction de pression. Le schéma numérique est adaptées aux écoulements faiblement compressibles et assure la positivité de la masse volumique et de l'énergie interne au niveau discret. Plusieurs cas tests numériques instationnaires ou stationnaires sont traités et montrent la capacité de l'approche intégrale à gérer des écoulements encombrés de tubes axiaux ou transverses à la direction de l'écoulement<br>The thesis issue is the modelling of fluid flows in congsted media by solid obstacles. The purpose is to design an integral approach reconciling the local and the component global scale for the numerical simulation of the coolant flow in the nuclear reactor components. The approcah affords the advantage of embedding the local and "component" representation scales in the same formalism, in ensuring the coherence between the two scales. This technique consists of a multidimensional integral formulation of the fluid flow governing equations allowing to naturally recover the CFD (Computational Fluid Dynamics) standard fluid approach when refining the mesh. The discretization is based on a time-implicit collocated finite volume numerical scheme using a pressure-correction algorithm. The scheme is relevant for weakly compressilbe flows and preserves the positivity of both the density and the internal energy, at the discrete level. Numerous unsteady or steady numerical tests are carried out and show the integral approach ability to simulate channel flows congested by axial or transversal rods

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7

Gokpi, Kossivi. "Modélisation et Simulation des Ecoulements Compressibles par la Méthode des Eléments Finis Galerkin Discontinus." Thesis, Pau, 2013. http://www.theses.fr/2013PAUU3005/document.

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L’objectif de ce travail de thèse est de proposer la Méthodes des éléments finis de Galerkin discontinus (DGFEM) à la discrétisation des équations compressibles de Navier-Stokes. Plusieurs challenges font l’objet de ce travail. Le premier aspect a consisté à montrer l’ordre de convergence optimal de la méthode DGFEM en utilisant les polynômes d’interpolation d’ordre élevé. Le deuxième aspect concerne l’implémentation de méthodes de ‘‘shock-catpuring’’ comme les limiteurs de pentes et les méthodes de viscosité artificielle pour supprimer les oscillations numériques engendrées par l’ordre élevé (lorsque des polynômes d’interpolation de degré p>0 sont utilisés) dans les écoulements transsoniques et supersoniques. Ensuite nous avons implémenté des estimateurs d’erreur a posteriori et des procédures d ’adaptation de maillages qui permettent d’augmenter la précision de la solution et la vitesse de convergence afin d’obtenir un gain de temps considérable. Finalement, nous avons montré la capacité de la méthode DG à donner des résultats corrects à faibles nombres de Mach. Lorsque le nombre de Mach est petit pour les écoulements compressibles à la limite de l’incompressible, la solution souffre généralement de convergence et de précision. Pour pallier ce problème généralement on procède au préconditionnement qui modifie les équations d’Euler. Dans notre cas, les équations ne sont pas modifiées. Dans ce travail, nous montrons la précision et la robustesse de méthode DG proposée avec un schéma en temps implicite de second ordre et des conditions de bords adéquats<br>The aim of this thesis is to deal with compressible Navier-Stokes flows discretized by Discontinuous Galerkin Finite Elements Methods. Several aspects has been considered. One is to show the optimal convergence of the DGFEM method when using high order polynomial. Second is to design shock-capturing methods such as slope limiters and artificial viscosity to suppress numerical oscillation occurring when p>0 schemes are used. Third aspect is to design an a posteriori error estimator for adaptive mesh refinement in order to optimize the mesh in the computational domain. And finally, we want to show the accuracy and the robustness of the DG method implemented when we reach very low mach numbers. Usually when simulating compressible flows at very low mach numbers at the limit of incompressible flows, there occurs many kind of problems such as accuracy and convergence of the solution. To be able to run low Mach number problems, there exists solution like preconditioning. This method usually modifies the Euler. Here the Euler equations are not modified and with a robust time scheme and good boundary conditions imposed one can have efficient and accurate results

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Singh, Manish Kumar. "LU-SGS Implicit Scheme For A Mesh-Less Euler Solver." Thesis, 2010. https://etd.iisc.ac.in/handle/2005/2397.

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Least Square Kinetic Upwind Method (LSKUM) belongs to the class of mesh-less method that solves compressible Euler equations of gas dynamics. LSKUM is kinetic theory based upwind scheme that operates on any cloud of points. Euler equations are derived from Boltzmann equation (of kinetic theory of gases) after taking suitable moments. The basic update scheme is formulated at Boltzmann level and mapped to Euler level by suitable moments. Mesh-less solvers need only cloud of points to solve the governing equations. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. Additionally blanking and de-blanking strategy along with overlay point cloud can be adapted in some applications like store separation to avoid regeneration of points. Naturally, the mesh-less solvers have advantage in tackling complex geometries and moving components over solvers that need grids. Conventionally, higher order accuracy for space derivative term is achieved by two step defect correction formula which is computationally expensive. The present solver uses low dissipation single step modified CIR (MCIR) scheme which is similar to first order LSKUM formulation and provides spatial accuracy closer to second order. The maximum time step taken to march solution in time is limited by stability criteria in case of explicit time integration procedure. Because of this, explicit scheme takes a large number of iterations to achieve convergence. The popular explicit time integration schemes like four stages Runge-Kutta (RK4) are slow in convergence due to this reason. The above problem can be overcome by using the implicit time integration procedure. The implicit schemes are unconditionally stable i.e. very large time steps can be used to accelerate the convergence. Also it offers superior robustness. The implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is very attractive due to its low numerical complexity, moderate memory requirement and unconditional stability for linear wave equation. Also this scheme is more efficient than explicit counterparts and can be implemented easily on parallel computers. It is based on the factorization of the implicit operator into three parts namely lower triangular matrix, upper triangular matrix and diagonal terms. The use of LU-SGS results in a matrix free implicit framework which is very economical as against other expensive procedures which necessarily involve matrix inversion. With implementation of the implicit LU-SGS scheme larger time steps can be used which in turn will reduce the computational time substantially. LU-SGS has been used widely for many Finite Volume Method based solvers. The split flux Jacobian formulation as proposed by Jameson is most widely used to make implicit procedure diagonally dominant. But this procedure when applied to mesh-less solvers leads to block diagonal matrix which again requires expensive inversion. In the present work LU-SGS procedure is adopted for mesh-less approach to retain diagonal dominancy and implemented in 2-D and 3-D solvers in matrix free framework. In order to assess the efficacy of the implicit procedure, both explicit and implicit 2-D solvers are tested on NACA 0012 airfoil for various flow conditions in subsonic and transonic regime. To study the performance of the solvers on different point distributions two types of the cloud of points, one unstructured distribution (4074 points) and another structured distribution (9600 points) have been used. The computed 2-D results are validated against NASA experimental data and AGARD test case. The density residual and lift coefficient convergence history is presented in detail. The maximum speed up obtained by use of implicit procedure as compared to explicit one is close to 6 and 14 for unstructured and structured point distributions respectively. The transonic flow over ONERA M6 wing is a classic test case for CFD validation because of simple geometry and complex flow. It has sweep angle of 30° and 15.6° at leading edge and trailing edge respectively. The taper ratio and aspect ratio of the wing are 0.562 and 3.8 respectively. At M∞=0.84 and α=3.06° lambda shock appear on the upper surface of the wing. 3¬D explicit and implicit solvers are tested on ONERA M6 wing. The computed pressure coefficients are compared with experiments at section of 20%, 44%, 65%, 80%, 90% and 95% of span length. The computed results are found to match very well with experiments. The speed up obtained from implicit procedure is over 7 for ONERA M6 wing. The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft use some form of the delta wing. To demonstrate the effectiveness of 3-D solver in handling control surfaces and small gaps, implicit 3-D code is used to compute flow past clipped delta wing with aileron deflection of 6° at M∞ = 0.9 and α = 1° and 3°. The leading edge backward sweep is 50.4°. The aileron is hinged from 56.5% semi-span to 82.9% of semi-span and at 80% of the local chord from leading edge. The computed results are validated with NASA experiments

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9

Singh, Manish Kumar. "LU-SGS Implicit Scheme For A Mesh-Less Euler Solver." Thesis, 2010. http://etd.iisc.ernet.in/handle/2005/2397.

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Least Square Kinetic Upwind Method (LSKUM) belongs to the class of mesh-less method that solves compressible Euler equations of gas dynamics. LSKUM is kinetic theory based upwind scheme that operates on any cloud of points. Euler equations are derived from Boltzmann equation (of kinetic theory of gases) after taking suitable moments. The basic update scheme is formulated at Boltzmann level and mapped to Euler level by suitable moments. Mesh-less solvers need only cloud of points to solve the governing equations. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. Additionally blanking and de-blanking strategy along with overlay point cloud can be adapted in some applications like store separation to avoid regeneration of points. Naturally, the mesh-less solvers have advantage in tackling complex geometries and moving components over solvers that need grids. Conventionally, higher order accuracy for space derivative term is achieved by two step defect correction formula which is computationally expensive. The present solver uses low dissipation single step modified CIR (MCIR) scheme which is similar to first order LSKUM formulation and provides spatial accuracy closer to second order. The maximum time step taken to march solution in time is limited by stability criteria in case of explicit time integration procedure. Because of this, explicit scheme takes a large number of iterations to achieve convergence. The popular explicit time integration schemes like four stages Runge-Kutta (RK4) are slow in convergence due to this reason. The above problem can be overcome by using the implicit time integration procedure. The implicit schemes are unconditionally stable i.e. very large time steps can be used to accelerate the convergence. Also it offers superior robustness. The implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is very attractive due to its low numerical complexity, moderate memory requirement and unconditional stability for linear wave equation. Also this scheme is more efficient than explicit counterparts and can be implemented easily on parallel computers. It is based on the factorization of the implicit operator into three parts namely lower triangular matrix, upper triangular matrix and diagonal terms. The use of LU-SGS results in a matrix free implicit framework which is very economical as against other expensive procedures which necessarily involve matrix inversion. With implementation of the implicit LU-SGS scheme larger time steps can be used which in turn will reduce the computational time substantially. LU-SGS has been used widely for many Finite Volume Method based solvers. The split flux Jacobian formulation as proposed by Jameson is most widely used to make implicit procedure diagonally dominant. But this procedure when applied to mesh-less solvers leads to block diagonal matrix which again requires expensive inversion. In the present work LU-SGS procedure is adopted for mesh-less approach to retain diagonal dominancy and implemented in 2-D and 3-D solvers in matrix free framework. In order to assess the efficacy of the implicit procedure, both explicit and implicit 2-D solvers are tested on NACA 0012 airfoil for various flow conditions in subsonic and transonic regime. To study the performance of the solvers on different point distributions two types of the cloud of points, one unstructured distribution (4074 points) and another structured distribution (9600 points) have been used. The computed 2-D results are validated against NASA experimental data and AGARD test case. The density residual and lift coefficient convergence history is presented in detail. The maximum speed up obtained by use of implicit procedure as compared to explicit one is close to 6 and 14 for unstructured and structured point distributions respectively. The transonic flow over ONERA M6 wing is a classic test case for CFD validation because of simple geometry and complex flow. It has sweep angle of 30° and 15.6° at leading edge and trailing edge respectively. The taper ratio and aspect ratio of the wing are 0.562 and 3.8 respectively. At M∞=0.84 and α=3.06° lambda shock appear on the upper surface of the wing. 3¬D explicit and implicit solvers are tested on ONERA M6 wing. The computed pressure coefficients are compared with experiments at section of 20%, 44%, 65%, 80%, 90% and 95% of span length. The computed results are found to match very well with experiments. The speed up obtained from implicit procedure is over 7 for ONERA M6 wing. The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft use some form of the delta wing. To demonstrate the effectiveness of 3-D solver in handling control surfaces and small gaps, implicit 3-D code is used to compute flow past clipped delta wing with aileron deflection of 6° at M∞ = 0.9 and α = 1° and 3°. The leading edge backward sweep is 50.4°. The aileron is hinged from 56.5% semi-span to 82.9% of semi-span and at 80% of the local chord from leading edge. The computed results are validated with NASA experiments

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10

Dlamini, Phumlani Goodwill. "Numerical simulation of finite-time blow-up in nonlinear ODEs, reaction-diffusion equations and VIDEs." Thesis, 2012. http://hdl.handle.net/10210/8054.

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M.Sc.<br>There have been an extensive study on solutions of differential equations modeling physical phenomena that blows up in finite time. The blow-up time often represents an important change in the properties of such models and hence it is very important to compute it as accurate as possible. In this work, an adaptive in time numerical method for computing blow-up solutions for nonlinear ODEs is introduced. The method is named implicit midpoint-implicit Euler method (IMIE) and is based on the implicit Euler and the implicit midpoint method. The method is used to compute blow-up time for different examples of ODEs, PDEs and VIDEs. The PDEs studied are reaction-diffusion equations whereby the method of lines is first used to discretize the equation in space to obtain a system of ODEs. Quadrature rules are used to approximate the integral in the VIDE to get a system of ODEs. The IMIE method is then used then to solve the system of ODEs. The results are compared to results obtained by the PECEIE method and Matlab solvers ode45 and ode15s. The results show that the IMIE method gives better results than the PECE-IE and ode15s and compares quite remarkably with the 4th order ode45 yet it is of order 1 with order 2 superconvergence at the mesh points.

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Books on the topic "Implicit Euler Method"

1

Blanco, Max. An implicit solution method for the Euler equations on unstructured triangular grids. National Library of Canada, 1995.

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Institute for Computer Applications in Science and Engineering., ed. Implicit schemes and parallel computing in unstructured grid CFD. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1995.

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Institute for Computer Applications in Science and Engineering., ed. Implicit schemes and parallel computing in unstructured grid CFD. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1995.

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Caughey, D. A. Multigrid calculation of three-dimensional turbomachinery flows. Fluid Dynamics and Aerodynamics Program, Sibley School of Mechanical and Aerospace Engineering, Cornell University, 1989.

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United States. National Aeronautics and Space Administration, ed. Multigrid calculation of three-dimensional turbomachinery flows. Fluid Dynamics and Aerodynamics Program, Sibley School of Mechanical and Aerospace Engineering, Cornell University, 1989.

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Explicit and implicit compact high-resolution shock-capturing methods for multidimensional Euler equations I, formulation. National Aeronautics and Space Administration, Ames Research Center, 1995.

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Rajeev, S. G. Finite Difference Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0014.

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This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.

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Book chapters on the topic "Implicit Euler Method"

1

Faragó, István. "Note on the Convergence of the Implicit Euler Method." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-41515-9_1.

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Sandu, Adrian, and Amik St-Cyr. "Stability Analysis of the Matrix-Free Linearly Implicit Euler Method." In Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35275-1_47.

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Sundnes, Joakim. "Stable Solvers for Stiff ODE Systems." In Solving Ordinary Differential Equations in Python. Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-46768-4_3.

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AbstractIn the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

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Ma, Wei-jun, and Qi-min Zhang. "Convergence of the Semi-implicit Euler Method for Stochastic Age-Dependent Population Equations with Markovian Switching." In Information Computing and Applications. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-16167-4_52.

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Song, Cheng, Longying Hu, and Haiyan Yuan. "The Mean Square Stability of Semi-implicit Euler Method for the Model of Technology Innovation Network." In Advances in Intelligent Systems and Computing. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00211-4_28.

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Hairer, Ernst, and Gerhard Wanner. "Euler Methods, Explicit, Implicit, Symplectic." In Encyclopedia of Applied and Computational Mathematics. Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_111.

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Walters, Robert W., James L. Thomas, and Bram Van Leer. "An implicit flux-split algorithm for the compressible Euler and Navier-Stokes equations." In Tenth International Conference on Numerical Methods in Fluid Dynamics. Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0041862.

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Elistratov, Andrei A., Dmitry V. Savin, and Olga B. Isaeva. "Complex Dynamics of the Implicit Maps Derived from Iteration of Newton and Euler Methods." In Communications in Computer and Information Science. Springer Nature Switzerland, 2022. http://dx.doi.org/10.1007/978-3-031-24145-1_3.

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Lecheler, S., and H. H. Frühauf. "A Fully Implicit 3-D Euler-Solver for Accurate and Fast Turbomachinery Flow Calculation." In Proceedings of the Ninth GAMM-Conference on Numerical Methods in Fluid Mechanics. Vieweg+Teubner Verlag, 1992. http://dx.doi.org/10.1007/978-3-663-13974-4_27.

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Rosli, Norhayati, Noor Amalina Nisa Ariffin, Yeak Su Hoe, and Arifah Bahar. "Stability Analysis of Explicit and Semi-implicit Euler Methods for Solving Stochastic Delay Differential Equations." In Proceedings of the Third International Conference on Computing, Mathematics and Statistics (iCMS2017). Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-13-7279-7_22.

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Conference papers on the topic "Implicit Euler Method"

1

Kim, MiHyeon, Juhyoung Park, and YoungBin Kim. "IM-BERT: Enhancing Robustness of BERT through the Implicit Euler Method." In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 2024. http://dx.doi.org/10.18653/v1/2024.emnlp-main.907.

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VASSBERG, JOHN. "A fast, implicit unstructured-mesh Euler method." In 10th Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics, 1992. http://dx.doi.org/10.2514/6.1992-2693.

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Chen, Shih H., and Anthony H. Eastland. "Implicit Euler Method for Three-Dimensional Turbomachinery Flow Calculation." In ASME 1993 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/93-gt-037.

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A compressible three-dimensional implicit Euler solution method for turbomachinery flows has been developed. The goal of the present study is to develop an efficient and reliable method that can be used to replace the semi-empirical, semi-analytical quasi-three-dimensional turbomachinery flow prediction method currently being used for multi-stage turbomachinery design at early design stages. Currently, a methodology has been developed based on an inviscid flow model (Euler solver) and tested on single blade rows for validation. The method presented here is derived from the Beam and Warming implicit approximate factorization (AF) finite difference algorithm. To avoid high frequency numerical instabilities associated with the use of central differencing schemes to obtain a spatial second order accuracy, a combined explicit and implicit artificial dissipation model is adopted. This model consists of a second order implicit dissipation and mixed second/fourth order explicit dissipation terms. A Cartesian coordinate H-grid generated by a three-dimensional interactive grid generator developed by Beach is used. Results for SSME High Pressure Fuel Turbine are presented and the comparison with experimental data is discussed. The use of the present implicit Euler method and the three-dimensional turbomachinery interactive grid generator shows that turnaround time could be as short as one day using a workstation. This allows the designers to explore optimal design configurations at minimum cost.

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CHEN, SHIH, and GEORGE PRUEGER. "Multistage turbomachinery flow solutions using three-dimensional implicit Euler method." In 29th Joint Propulsion Conference and Exhibit. American Institute of Aeronautics and Astronautics, 1993. http://dx.doi.org/10.2514/6.1993-2382.

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WHITAKER, DAVID. "Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method." In 11th Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics, 1993. http://dx.doi.org/10.2514/6.1993-3337.

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Zhang, Yining, Haochun Zhang, Yang Su, and Guangbo Zhao. "A Comparative Study of 10 Different Methods on Numerical Solving of Point Reactor Neutron Kinetics Equations." In 2017 25th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/icone25-67275.

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Point reactor neutron kinetics equations describe the time dependent neutron density variation in a nuclear reactor core. These equations are widely applied to nuclear system numerical simulation and nuclear power plant operational control. This paper analyses the characteristics of 10 different basic or normal methods to solve the point reactor neutron kinetics equations. These methods are: explicit and implicit Euler method, explicit and implicit four order Runge-Kutta method, Taylor polynomial method, power series method, decoupling method, end point floating method, Hermite method, Gear method. Three different types of step reactivity values are introduced respectively at initial time when point reactor neutron kinetics equations are calculated using different methods, which are positive reactivity, negative reactivity and higher positive reactivity. The calculation results show that (i) minor relative error can be gain after three types of step reactivity are introduced, when explicit or implicit four order Runge-Kutta method, Taylor polynomial method, power series method, end point floating method or Hermite method is taken. These methods which are mentioned above are appropriate for solving point reactor neutron kinetics equations. (ii) the relative error of decoupling method is large, under the calculation condition of this paper. When a higher reactivity is introduced, the calculation of decoupling method cannot be convergence. (iii) after three types of step reactivity are introduced respectively, the relative error of implicit Euler method is higher than any other method except decoupling method. The third highest is Gear method. (iv) when the higher reactivity is introduced, the relative error of explicit and implicit Euler method are almost coincident, and higher than any other methods obviously. (v) 4 methods are suitable for solution on these given conditions, which are implicit Runge-Kutta method, Taylor polynomial method, power series method and end point floating method, considering both the accuracy and stiffness.

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Varnhorn, W. "An Approximation Method for Non-Stationary Stokes Flow." In Topical Problems of Fluid Mechanics 2023. Institute of Thermomechanics of the Czech Academy of Sciences; CTU in Prague Faculty of Mech. Engineering Dept. Tech. Mathematics, 2023. http://dx.doi.org/10.14311/tpfm.2023.027.

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We consider a rst order implicit time stepping procedure (Euler scheme) for the non-stationary Stokes equations in smoothly bounded domains of R³. Using energy estimates we can prove optimal convergence properties in the Sobolev spaces Hᵐ(G) (m = 0, 1, 2) uniformly in time, provided that the solution of the Stokes equations has a certain degree of regularity.

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Tu, Tianxiong, Guoping Wang, Xiaoting Rui, Jianshu Zhang, and Xiangzhen Zhou. "Implicit Algorithm for Riccati Transfer Matrix Method for Multibody Systems." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-85675.

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Rui method, namely the transfer matrix method for multibody systems (MSTMM) is a new and efficient method for multibody system dynamics (MSD) for its features as follows: without global dynamics equations of the system, high programming, low order of system matrix and high computational speed. Riccati transfer matrix method for multibody systems was developed by introducing Riccati transformation in MSTMM, for improving numerical stability of MSTMM. In this paper, based on Riccati MSTMM, applying the thought of direct differentiation method, by differentiation of Riccati transfer equations of rigid bodies and joints, generalized acceleration and its differentiation can be obtained. Combined with Backward Euler algorithm, implicit algorithm for Riccati MSTMM is proposed in this paper. The formulation and computing procedure of the method are presented. The numerical examples show that results obtained by first order accurate implicit algorithm proposed in the paper and the fourth order accurate Runge-Kutta method have good agreement, which indicates that this implicit method is more numerical stability than explicit algorithm with the same order accurate. The implicit algorithm for Riccati MSTMM can be used for improving the computational accuracy of multibody system dynamics.

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Hou, Wenguo, Peter X. Liu, Minhua Zheng, and Shichao Liu. "A New FEM-based Brain Tissue Model for Neurosurgical Simulation Using the Optimization Implicit Euler Method." In 2018 37th Chinese Control Conference (CCC). IEEE, 2018. http://dx.doi.org/10.23919/chicc.2018.8484193.

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Zhao, Aihong A. (Rachel), and C. L. Chow. "Computational Algorithms for a Damage-Coupled Cyclic Viscoplasticity Material Model." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84930.

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The primary objective of the investigation is to develop efficient and robust computational schemes for a damage-coupled cyclic thermoviscoplasticity model for solder material. Three constitutive integration algorithms, Euler, modified Euler and semi-implicit algorithm for the model are examined. The three algorithms for the model are coded in the commercial finite element (FE) code ABAQUS (version 6.21) via the user-defined material subroutine UMAT. Two single-step algorithms of the substep scheme are applied for the modified Euler algorithm to control the error in the integration of constitutive laws. A semi-empirical formulation is established for an adaptive time stepping algorithm that is based on the Euler algorithm. Single-element, miniature specimen and notched specimen simulations have been conducted to compare with the test results, which include monotonic tensile, creep and fatigue tests of 63Sn-37Pb solder. It is observed that the explicit algorithm consistently requires much less CPU time than others. The modified Euler algorithm has shown on the other hand to be not only efficient but also accurate. The semi-implicit algorithm yields accurate solution. It is worth noting that the method is also effective when an appropriate integration scheme is chosen.

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