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Journal articles on the topic 'Second-order backward time scheme'

Author: Grafiati

Published: 19 January 2023

Last updated: 31 July 2025

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1

Vabishchevich, Petr N. "Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems." Computational Methods in Applied Mathematics 17, no. 2 (2017): 323–35. http://dx.doi.org/10.1515/cmam-2016-0038.

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AbstractSchemes with the second-order approximation in time are considered for numerically solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Padé polynomial approximation is unconditionally stable. It demonstrates good asymptotic properties in time and provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic (SM) stability). In fact, the only drawback of this scheme is the necessity to solve an equation with an operator polynomial of second degree at each t

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2

Bokanowski, Olivier, Athena Picarelli, and Christoph Reisinger. "High-order filtered schemes for time-dependent second order HJB equations." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 1 (2018): 69–97. http://dx.doi.org/10.1051/m2an/2017039.

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In this paper, we present and analyse a class of “filtered” numerical schemes for second order Hamilton–Jacobi–Bellman (HJB) equations. Our approach follows the ideas recently introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal. 51 (2013) 423–444, and more recently applied by other authors to stationary or time-dependent first order Hamilton–Jacobi equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the class

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3

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

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

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4

Sahu, Subal Ranjan, and Jugal Mohapatra. "Numerical investigation of time delay parabolic differential equation involving two small parameters." Engineering Computations 38, no. 6 (2021): 2882–99. http://dx.doi.org/10.1108/ec-07-2020-0369.

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Purpose The purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP). Design/methodology/approach To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization. Findings It is observed that the proposed method con

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BERNARDI, CHRISTINE, and ENDRE SÜLI. "TIME AND SPACE ADAPTIVITY FOR THE SECOND-ORDER WAVE EQUATION." Mathematical Models and Methods in Applied Sciences 15, no. 02 (2005): 199–225. http://dx.doi.org/10.1142/s0218202505000339.

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The aim of this paper is to show that, for a linear second-order hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise affine finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.

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6

Ravindran, S. S. "An Extrapolated Second Order Backward Difference Time-Stepping Scheme for the Magnetohydrodynamics System." Numerical Functional Analysis and Optimization 37, no. 8 (2016): 990–1020. http://dx.doi.org/10.1080/01630563.2016.1181651.

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7

Wang, Danxia, Ni Miao, and Jing Liu. "A second-order numerical scheme for the Ericksen-Leslie equation." AIMS Mathematics 7, no. 9 (2022): 15834–53. http://dx.doi.org/10.3934/math.2022867.

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<abstract><p>In this paper, we consider a finite element approximation for the Ericksen-Leslie model of nematic liquid crystal. Based on a saddle-point formulation of the director vector, a second-order backward differentiation formula (BDF) numerical scheme is proposed, where a pressure-correction strategy is used to decouple the computation of the pressure from that of the velocity. Designing this scheme leads to solving a linear system at each time step. Furthermore, via implementing rigorous theoretical analysis, we prove that the proposed scheme enjoys the energy dissipation l

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8

Park, Sang-Hun, and Tae-Young Lee. "High-Order Time-Integration Schemes with Explicit Time-Splitting Methods." Monthly Weather Review 137, no. 11 (2009): 4047–60. http://dx.doi.org/10.1175/2009mwr2885.1.

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Abstract New high-order time-integration schemes for fully elastic models are presented. The new schemes, formulated using the Richardson extrapolation that employs leapfrog-type schemes, can give a good performance for linear model problems and ensure overall stability when they are combined with a forward–backward scheme for fast waves. The new and existing schemes show differences in the order of accuracy. Thus, they can be useful for investigating the impacts of time-integration scheme accuracy on the performance of numerical models. The high-order schemes are found to play an important ro

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9

Chibuisi, C., B. O. Osu, U. W. Sirisena, K. Uchendu, and C. Granados. "The Computational Solution of First Order Delay Differential Equations Using Second Derivative Block Backward Differentiation Formulae." International Journal of Mathematical Analysis and Optimization: Theory and Applications 7, no. 2 (2022): 88–106. http://dx.doi.org/10.52968/28304669.

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In this paper, we implemented second derivative block backward differentiation formulae methods in solving first order delay differential equations without the application of interpolation methods in investigating the delay argument. The delay argument was evaluated using a suitable idea of sequence which we incorporated into some first order delay differential equations before its numerical evaluations. The construction of the continuous expressions of these of block methods was executed through the use of second derivative backward differentiation formulae method on the bases of linear multi

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10

He, Haiyan, Kaijie Liang, and Baoli Yin. "A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation." International Journal of Modeling, Simulation, and Scientific Computing 10, no. 01 (2019): 1941005. http://dx.doi.org/10.1142/s1793962319410058.

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In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement

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11

Hou, Yaxin, Ruihan Feng, Yang Liu, Hong Li, and Wei Gao. "A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system." International Journal of Modeling, Simulation, and Scientific Computing 08, no. 01 (2017): 1750012. http://dx.doi.org/10.1142/s179396231750012x.

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In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

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12

Zhang, Na, Weihua Deng, and Yujiang Wu. "Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation." Advances in Applied Mathematics and Mechanics 4, no. 04 (2012): 496–518. http://dx.doi.org/10.4208/aamm.10-m1210.

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AbstractWe present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on theL1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and conv

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Zhao, Yanzi, and Xinlong Feng. "Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods." Entropy 24, no. 10 (2022): 1338. http://dx.doi.org/10.3390/e24101338.

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In this paper, an effective numerical algorithm for the Stokes equation of a curved surface is presented and analyzed. The velocity field was decoupled from the pressure by the standard velocity correction projection method, and the penalty term was introduced to make the velocity satisfy the tangential condition. The first-order backward Euler scheme and second-order BDF scheme are used to discretize the time separately, and the stability of the two schemes is analyzed. The mixed finite element pair (P2,P1) is applied to discretization of space. Finally, numerical examples are given to verify

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14

Weng, Zhifeng, Langyang Huang, and Rong Wu. "Numerical Approximation of the Space Fractional Cahn-Hilliard Equation." Mathematical Problems in Engineering 2019 (April 1, 2019): 1–10. http://dx.doi.org/10.1155/2019/3163702.

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In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, i

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15

Nong, Lijuan, An Chen, and Jianxiong Cao. "Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data." Mathematical Modelling of Natural Phenomena 16 (2021): 12. http://dx.doi.org/10.1051/mmnp/2021007.

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In this paper, we consider a two-term time-fractional diffusion-wave equation which involves the fractional orders α ∈ (1, 2) and β ∈ (0, 1), respectively. By using piecewise linear Galerkin finite element method in space and convolution quadrature based on second-order backward difference method in time, we obtain a robust fully discrete scheme. Error estimates for semidiscrete and fully discrete schemes are established with respect to nonsmooth data. Numerical experiments for two-dimensional problems are provided to illustrate the efficiency of the method and conform the theoretical results.

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16

Meidner, Dominik, and Thomas Richter. "Goal-Oriented Error Estimation for the Fractional Step Theta Scheme." Computational Methods in Applied Mathematics 14, no. 2 (2014): 203–30. http://dx.doi.org/10.1515/cmam-2014-0002.

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Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properti

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17

Wang, Deng, Yang Liu, Hong Li, and Zhichao Fang. "Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations." Fractal and Fractional 6, no. 4 (2022): 201. http://dx.doi.org/10.3390/fractalfract6040201.

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In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mi

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18

He, Li-Ping, and Minxin He. "Parareal in Time Simulation Of Morphological Transformation in Cubic Alloys with Spatially Dependent Composition." Communications in Computational Physics 11, no. 5 (2012): 1697–717. http://dx.doi.org/10.4208/cicp.110310.090911a.

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AbstractIn this paper, a reduced morphological transformation model with spatially dependent composition and elastic modulus is considered. The parareal in time al-gorithm introduced by Lions et al. is developed for longer-time simulation. The fine solver is based on a second-order scheme in reciprocal space, and the coarse solver is based on a multi-model backward Euler scheme, which is fast and less expensive. Numerical simulations concerning the composition with a random noise and a discontinuous curve are performed. Some microstructure characteristics at very low temperature are obtained b

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19

Ma, Ying, and Lizhen Chen. "Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation." Fractal and Fractional 6, no. 8 (2022): 439. http://dx.doi.org/10.3390/fractalfract6080439.

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We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have min{2−α,2−β}-order convergence in time and spectral accuracy in space for smooth solutions, where α,β are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness

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20

LI, Juan. "Energy Stable BDF2-SAV Scheme on Variable Grids for the Epitaxial Thin Film Growth Models." Wuhan University Journal of Natural Sciences 29, no. 6 (2024): 517–22. https://doi.org/10.1051/wujns/2024296517.

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The second-order backward differential formula (BDF2) and the scalar auxiliary variable (SAV) approach are applied to construct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models. Under the step-ratio condition 0[see formula in PDF]4.864, the modified energy dissipation law is proven at the discrete levels with regardless of time step size. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.

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21

KUMAR, B. V. RATHISH, and MANI MEHRA. "A WAVELET-TAYLOR GALERKIN METHOD FOR PARABOLIC AND HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS." International Journal of Computational Methods 02, no. 01 (2005): 75–97. http://dx.doi.org/10.1142/s0219876205000375.

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In this study a set of new space and time accurate numerical methods based on different time marching schemes such as Euler, leap-frog and Crank-Nicolson for partial differential equations of the form [Formula: see text], where ℒ is linear differential operator and [Formula: see text] is a nonlinear function, are proposed. To produce accurate temporal differencing, the method employs forward/backward time Taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation. This yields a generalized time discretized

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22

Wang, Hongjian. "Numerical Analysis of a BDF2 finite Element Scheme for Navier-Stokes-Omega Model." Scholars Journal of Physics, Mathematics and Statistics 11, no. 09 (2024): 120–31. http://dx.doi.org/10.36347/sjpms.2024.v11i09.003.

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In this paper, we study a second-order backward difference formula (BDF2) scheme for the Navier-Stokes-omega (NS-omega) model. By employing the stabilization scheme for space discretization and the BDF2 method for time discretization of NS-omega model, we obtain the fully discrete approximation of them. The paper provides an analysis of the unconditional stability and convergence of the approximate solutions. Furthermore, the numerical experiments are conducted to validate the theoretical findings and demonstrate the efficiency of the proposed scheme.

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23

Ray, Saurya Ranjan, and Josef Ballmann. "Backward Difference Scheme for Simulating Unsteady Compressible Flow on Deforming Mesh in an Implicit Adaptive Solver." Applied Mechanics and Materials 598 (July 2014): 493–97. http://dx.doi.org/10.4028/www.scientific.net/amm.598.493.

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The paper describes the derivation of the numerical formulation of a second order time accurate and Geometrically Conservative Backward Difference Scheme (BDF) for transient flow simulation of Arbitrary Lagrangian Eulerian (ALE) problems using the control volume approach. The required modification to implement the scheme in an implicit adaptive flow solver is explained. The accuracy and robustness of the current formulation is demonstrated by simulating unsteady flow field over a sinusoidally pitching NACA0012 airfoil with larger allowable timestep in comparison to an existing Mid-point scheme

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24

Izydorczyk, Lucas, Nadia Oudjane, and Francesco Russo. "A fully backward representation of semilinear PDEs applied to the control of thermostatic loads in power systems." Monte Carlo Methods and Applications 27, no. 4 (2021): 347–71. http://dx.doi.org/10.1515/mcma-2021-2095.

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Abstract We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, the grid is generated adaptively in the areas of interest, and second, there is no need to store the entire grid. The performances of this technique are compared in simulations to the traditional Monte-Carlo forward-backward approach on a cont

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25

Ravindran, S. S. "A second-order backward difference time-stepping scheme for penalized Navier-Stokes equations modeling filtration through porous media." Numerical Methods for Partial Differential Equations 32, no. 2 (2015): 681–705. http://dx.doi.org/10.1002/num.22029.

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26

Wu, Longyuan, Xinlong Feng, and Yinnian He. "Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces." Entropy 25, no. 12 (2023): 1631. http://dx.doi.org/10.3390/e25121631.

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We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expa

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27

Yang, Yining, Cao Wen, Yang Liu, Hong Li, and Jinfeng Wang. "Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model." Communications in Analysis and Mechanics 16, no. 1 (2024): 24–52. http://dx.doi.org/10.3934/cam.2024002.

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<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was appli

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28

Huang, Yu-Yun, Xian-Ming Gu, Yi Gong, Hu Li, Yong-Liang Zhao, and Bruno Carpentieri. "A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations." Fractal and Fractional 5, no. 4 (2021): 230. http://dx.doi.org/10.3390/fractalfract5040230.

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In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on

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YAO, JIANYAO, G. R. LIU, DONG QIAN, CHUNG-LUNG CHEN, and GEORGE X. XU. "A MOVING-MESH GRADIENT SMOOTHING METHOD FOR COMPRESSIBLE CFD PROBLEMS." Mathematical Models and Methods in Applied Sciences 23, no. 02 (2013): 273–305. http://dx.doi.org/10.1142/s0218202513400046.

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A computational fluid dynamics (CFD) solver based on the gradient smoothing method (GSM) with moving mesh enabled is presented in this paper. The GSM uses unstructured meshes which could be generated and remeshed easily. The spatial derivatives of field variables at nodes and midpoints of cell edges are calculated using the gradient smoothing operations. The presented GSM codes use second-order Roes upwind flux difference splitting method and second-order 3-level backward differencing scheme for the compressible Navier–Stokes equations with moving mesh, and the second-order of accuracy for bot

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Liao, Cheng, Danxia Wang, and Haifeng Zhang. "The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model." Mathematics 12, no. 5 (2024): 672. http://dx.doi.org/10.3390/math12050672.

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In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed

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31

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

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

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Čiegis, Raimondas, and Remigijus Čiegis. "Numerical algorithms for one parabolic-elliptic problem." Lietuvos matematikos rinkinys 43 (December 22, 2003): 581–85. http://dx.doi.org/10.15388/lmr.2003.32532.

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In this paper we solve numerically a parabolic-elliptic problem. Two finite difference schemes are proposed. The first scheme is a modification of the backward Euler algorithm and it requires to solve an elliptic problem at each time step. The spectral estimates of the obtained matrix are presented. The second scheme is a modification of the stability-correction scheme. This scheme is used as a classical splitting scheme in the parabolic region of the problem definition and as a new iterative algorithm in the elliptic part of the problem. We prove the convergence of the proposed scheme.

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ITKIN, ANDREY. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs." International Journal of Theoretical and Applied Finance 18, no. 05 (2015): 1550031. http://dx.doi.org/10.1142/s0219024915500314.

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This paper is dedicated to the construction of high order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreassen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility (LSV) model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models wit

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Hernández-Sánchez, Mónica, Francisco-Shu Kitaura, Metin Ata, and Claudio Dalla Vecchia. "Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis." Monthly Notices of the Royal Astronomical Society 502, no. 3 (2021): 3976–92. http://dx.doi.org/10.1093/mnras/stab123.

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ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-

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Yin, Baoli, Guoyu Zhang, Yang Liu, and Hong Li. "The Construction of High-Order Robust Theta Methods with Applications in Subdiffusion Models." Fractal and Fractional 6, no. 8 (2022): 417. http://dx.doi.org/10.3390/fractalfract6080417.

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An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter θ to approximate fractional calculus operators. In contrast to the known θ methods obtained by polynomial-type transformations, our exponential-type θ methods take the advantage of the fact that they have no restrictions in theory on the range of θ such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain t

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Verma, Amit Kumar, Mukesh Kumar Rawani, and Ravi P. Agarwal. "A High-Order Weakly L-Stable Time Integration Scheme with an Application to Burgers’ Equation." Computation 8, no. 3 (2020): 72. http://dx.doi.org/10.3390/computation8030072.

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In this paper, we propose a 7th order weakly L-stable time integration scheme. In the process of derivation of the scheme, we use explicit backward Taylor’s polynomial approximation of sixth-order and Hermite interpolation polynomial approximation of fifth order. We apply this formula in the vector form in order to solve Burger’s equation, which is a simplified form of Navier-Stokes equation. The literature survey reveals that several methods fail to capture the solutions in the presence of inconsistency and for small values of viscosity, e.g., 10−3, whereas the present scheme produces highly

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Jiang, Nan. "A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations." Numerical Methods for Partial Differential Equations 33, no. 1 (2016): 34–61. http://dx.doi.org/10.1002/num.22070.

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Variansyah, Ilham, Edward W. Larsen, and William R. Martin. "A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS." EPJ Web of Conferences 247 (2021): 03024. http://dx.doi.org/10.1051/epjconf/202124703024.

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A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourie

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

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

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

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

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Nawaz, Yasir, Muhammad Shoaib Arif, Muavia Mansoor, Kamaleldin Abodayeh, and Amani S. Baazeem. "Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source." Fractal and Fractional 8, no. 5 (2024): 276. http://dx.doi.org/10.3390/fractalfract8050276.

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An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized b

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Tian, Shifang, Xiaowei Liu, and Ran An. "A Higher-Order Finite Difference Scheme for Singularly Perturbed Parabolic Problem." Mathematical Problems in Engineering 2021 (August 3, 2021): 1–11. http://dx.doi.org/10.1155/2021/9941692.

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In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.

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Oishi, Cassio M., José A. Cuminato, Valdemir G. Ferreira, et al. "A Stable Semi-Implicit Method for Free Surface Flows." Journal of Applied Mechanics 73, no. 6 (2005): 940–47. http://dx.doi.org/10.1115/1.2173672.

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The present work is concerned with a semi-implicit modification of the GENSMAC method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in primitive variables formulation with a free surface. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The viscous terms are treated by the implicit backward method in time and a centered second order method in space, and the nonlinear convection terms are explicitly approximated by the high order upwind variabl

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Durst, F., and J. C. F. Pereira. "Time-Dependent Laminar Backward-Facing Step Flow in a Two-Dimensional Duct." Journal of Fluids Engineering 110, no. 3 (1988): 289–96. http://dx.doi.org/10.1115/1.3243547.

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This paper presents results of numerical studies of the impulsively starting backward-facing step flow with the step being mounted in a plane, two-dimensional duct. Results are presented for Reynolds numbers of Re = 10; 368 and 648 and for the last two Reynolds numbers comparisons are given between experimental and numerical results obtained for the final steady state flow conditions. In the computational scheme, the convective terms in the momentum equations are approximated by a 13-point quadratic upstream weighted finite-difference scheme and a fully implicit first order forward differencin

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Guzel, Goktan, and Ilteris Koc. "Time-Accurate Flow Simulations Using a Finite-Volume Based Lattice Boltzmann Flow Solver with Dual Time Stepping Scheme." International Journal of Computational Methods 13, no. 06 (2016): 1650035. http://dx.doi.org/10.1142/s0219876216500353.

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In this study, the Lattice Boltzmann Method (LBM) is implemented through a finite-volume approach to perform 2D, incompressible, and time-accurate fluid flow analyses on structured grids. Compared to the standard LBM (the so-called stream and collide scheme), the finite-volume approach followed in this study necessitates more computational effort, but the major limitations of the former on grid uniformity and Courant–Friedrichs–Lewy (CFL) number that is to be one are removed. Even though these improvements pave the way for the possibility of solving more practical fluid flow problems with the

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Aazizi, Soufiane. "Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes." Advances in Applied Probability 45, no. 3 (2013): 791–821. http://dx.doi.org/10.1239/aap/1377868539.

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We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain the Lp-Hölder continuity of the solution of the FBSDEBP and

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Aazizi, Soufiane. "Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes." Advances in Applied Probability 45, no. 03 (2013): 791–821. http://dx.doi.org/10.1017/s0001867800006583.

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We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain theLp-Hölder continuity of the solution of the FBSDEBP and

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Yang, Jie, and Weidong Zhao. "Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation." East Asian Journal on Applied Mathematics 5, no. 4 (2015): 387–404. http://dx.doi.org/10.4208/eajam.280515.211015a.

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AbstractConvergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

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Bello, Abdullahi, Garba Ismail Danbaba, and Usman Garba. "CONVERGENCE TEST FOR THE EXTENDED 3 - POINT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR INTEGRATING STIFF IVP." FUDMA JOURNAL OF SCIENCES 7, no. 4 (2023): 103–12. http://dx.doi.org/10.33003/fjs-2023-0704-1906.

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In this work, a new scheme is generated from the extended 3–point super class of block backward differentiation formula for integrating stiff IVP and the proposed method is subjected to convergence test. The proposed scheme is found to be zero stable, consistent and of order 5. Thus, possess all the required criteria for convergence. The scheme can approximate the values of three points at a time per integration step. The scheme maintained the same technique of co-opting a stability control parameter () in the formula and by adjusting its value within the interval , more A-stabled schemes can

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Decaix, Jean, Andres Müller, Arthur Favrel, François Avellan, and Cécile Münch-Alligné. "Investigation of the Time Resolution Set Up Used to Compute the Full Load Vortex Rope in a Francis Turbine." Applied Sciences 11, no. 3 (2021): 1168. http://dx.doi.org/10.3390/app11031168.

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The flow in a Francis turbine at full load is characterised by the development of an axial vortex rope in the draft tube. The vortex rope often promotes cavitation if the turbine is operated at a sufficiently low Thoma number. Furthermore, the vortex rope can evolve from a stable to an unstable behaviour. For CFD, such a flow is a challenge since it requires solving an unsteady cavitating flow including rotor/stator interfaces. Usually, the numerical investigations focus on the cavitation model or the turbulence model. In the present works, attention is paid to the strategy used for the time i

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