Статті в журналах: "Second-order backward time scheme"

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Vabishchevich, Petr N. "Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 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 time level. We consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree. Such computational implementations occur, for example, if we apply the fully implicit two-level scheme (the backward Euler scheme). A three-level modification of the SM-stable scheme is proposed. Its unconditional stability is established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm involves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage and the following correction of the approximate solution using a factorized operator. The SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work, a model problem is solved numerically.

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 (January 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 classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by “filtering” them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.

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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 (February 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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Sahu, Subal Ranjan, and Jugal Mohapatra. "Numerical investigation of time delay parabolic differential equation involving two small parameters." Engineering Computations 38, no. 6 (January 20, 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 converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Originality/value This paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. 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. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.

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Ravindran, S. S. "An Extrapolated Second Order Backward Difference Time-Stepping Scheme for the Magnetohydrodynamics System." Numerical Functional Analysis and Optimization 37, no. 8 (May 4, 2016): 990–1020. http://dx.doi.org/10.1080/01630563.2016.1181651.

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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 law. Some numerical simulations are also performed to demonstrate the accuracy of the proposed scheme.</p></abstract>

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Park, Sang-Hun, and Tae-Young Lee. "High-Order Time-Integration Schemes with Explicit Time-Splitting Methods." Monthly Weather Review 137, no. 11 (November 1, 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 role in the improvement of high-resolution simulations, according to idealized tests. The new schemes are less efficient than other well-known schemes at moderate spatial resolutions. However, the new schemes can be more efficient than the existing schemes when the resolution becomes very high.

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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 (March 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 multistep collocation approach using matrix inversion method to derive the discrete schemes. After the numerical experiments, the new proposed method was observed to be convergent, stable and less time consuming. From the numerical solutions obtained, the scheme for step number k = 4 performed better in terms of accuracy than that of the schemes for step numbers k = 3 and 2 when compared with other existing methods.

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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 (February 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 the finite element method for the spatial approximation. Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained. Numerical experiments are carried out to demonstrate our theoretical analysis.

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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 (January 10, 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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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 (August 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 convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

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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 (September 23, 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 the accuracy and effectiveness of the proposed method.

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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, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

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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 (April 2, 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 mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results.

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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Meidner, Dominik, and Thomas Richter. "Goal-Oriented Error Estimation for the Fractional Step Theta Scheme." Computational Methods in Applied Mathematics 14, no. 2 (April 1, 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 properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

17

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 (May 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 by a variable temperature technique.

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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 (August 11, 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 of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.

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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 (November 2, 2015): 681–705. http://dx.doi.org/10.1002/num.22029.

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20

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 (March 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 scheme which is approximated in space by Galerkin method. The compactly supported orthogonal wavelet bases developed by Daubechies are used in Galerkin scheme. This new wavelet-Taylor Galerkin approach is successively applied to heat equation, convection equation and inviscid Burgers' equation.

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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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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 (October 21, 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 control problem of thermostatic loads.

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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 (November 18, 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 preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

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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 (January 8, 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 both the spatial and temporal discretization is ensured. The spatial discretization accuracy is verified using the method of manufactured solutions (MMS) on both structured and unstructured triangle meshes, and the results show that the observed order of accuracy achieves 2 even when highly distorted meshes are used. The temporal discretization accuracy is verified using the results with different time step lengths, and second-order accuracy is also obtained. Therefore, it is confirmed that the proposed GSM-CFD solver is a uniform second-order scheme.

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Jebens, Stefan, Oswald Knoth, and Rüdiger Weiner. "Explicit Two-Step Peer Methods for the Compressible Euler Equations." Monthly Weather Review 137, no. 7 (July 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 Runge–Kutta method by Wicker and Skamarock shows the potential of the class of peer methods with respect to computational efficiency, stability, and accuracy.

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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 (May 31, 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 Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed.

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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 (July 28, 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 with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.

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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 (July 29, 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 the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small α and can naturally resolve the initial singularity provided θ=−12, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests.

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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 (January 21, 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-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

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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 (August 9, 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 accurate results. To check the effectiveness of the scheme, we examine it over six test problems and generate several tables and figures. All of the calculations are executed with the help of Mathematica 11.3. The stability and convergence of the scheme are also discussed.

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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 the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under theL2norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.

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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 (September 25, 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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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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Führer, Thomas, Norbert Heuer, and Jhuma Sen Gupta. "A Time-Stepping DPG Scheme for the Heat Equation." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 2017): 237–52. http://dx.doi.org/10.1515/cmam-2016-0037.

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AbstractWe introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

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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 (September 1, 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 differencing scheme is used to discretize the temporal derivatives. The computations show that for the higher Reynolds numbers, the flow starts to separate on the lower and upper corners of the step yielding two disconnected recirculating flow regions for some time after the flow has been impulsively started. As time progresses, these two separated flow regions connect up and a single recirculating flow region emerges. This separated flow region stays attached to the step, grows in size and approaches, for the time t → ∞, the dimensions measured and predicted for the separation region for steady laminar backward-facing flow. For the Reynolds number Re = 10 the separation starts at the bottom of the backward-facing step and the separation region enlarges with time until the steady state flow pattern is reached. At the channel wall opposite to the step and for Reynolds number Re = 368, a separated flow region is observed and it is shown to occur for some finite time period of the developing, impulsively started backward-facing step flow. Its dimensions change with time and reduce to zero before the steady state flow pattern is reached. For the higher Reynolds number Re = 648, the secondary separated flow region opposite to the wall is also present and it is shown to remain present for t → ∞. Two kinds of the inlet conditions were considered; the inlet mean flow was assumed to be constant in a first study and was assumed to increase with time in a second one. The predicted flow field for t → ∞ turned out to be identical for both cases. They were also identical to the flow field predicted for steady, backward-facing step flow using the same numerical grid as for the time-dependent predictions.

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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 (November 2, 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 LBM, time-accurate simulations are still restricted due to the stability criteria dictated by high-aspect ratio grid cells that are usually required for adequate resolution of boundary layers and the stiffness due to the nature of the equation that are being solved. To overcome this limitation, a Dual Time Stepping (DTS) scheme, which iterates the solution in pseudo time using an Implicit-Explicit (IMEX) Runge–Kutta method while advancing the solution in physical time with an explicit scheme (backward difference formula), is developed and implemented. The accuracy of the resulting flow solver is evaluated using benchmark flow problems and overall second-order accuracy is demonstrated.

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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 (September 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 we construct two numerical schemes for this FBSDEBP. Based on the Lp-Hölder estimate, we prove the convergence of the scheme when the number of time steps n goes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

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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 (September 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 we construct two numerical schemes for this FBSDEBP. Based on theLp-Hölder estimate, we prove the convergence of the scheme when the number of time stepsngoes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

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Oishi, Cassio M., José A. Cuminato, Valdemir G. Ferreira, Murilo F. Tomé, Antonio Castelo, Norberto Mangiavacchi, and Sean McKee. "A Stable Semi-Implicit Method for Free Surface Flows." Journal of Applied Mechanics 73, no. 6 (December 30, 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 variable-order nonoscillatory scheme method in space. The boundary conditions at the free surface couple the otherwise segregated velocity and pressure fields. The present work proposes a method that allows the segregated solution of free surface flow problems to be computed by semi-implicit schemes that preserve the stability conditions of the related coupled semi-implicit scheme. The numerical method is applied to both the simulation of free surface and to confined flows. The numerical results demonstrate that the present technique eliminates the parabolic stability restriction required by the original explicit GENSMAC method, and also found in segregated semi-implicit methods with time-lagged boundary conditions. For low Reynolds number flows, the method is robust and very efficient when compared to the original GENSMAC method.

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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 (November 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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Rezaiee-Pajand, M., and S. R. Sarafrazi. "A Mixed and Multi-Step Higher-Order Implicit Time Integration Family." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 10 (April 22, 2010): 2097–108. http://dx.doi.org/10.1243/09544062jmes2093.

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This article develops a new time integration family for second-order dynamic equations. A combination of the trapezoidal rule and higher-order Newton backward extrapolation functions are utilized in the formulation. Five members of the suggested family are extensively studied in this article. Most members of the presented time integration family are new. The stability and accuracy of the proposed time integration schemes are investigated by solving some benchmark problems. Numerical results are checked and compared with well-known strategies. The findings of the article show the efficiency, accuracy and robustness of the suggested techniques.

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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 (January 27, 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 integration. The vortex rope considered is an unstable cavitating one that develops downstream the runner. The vortex rope shows a periodic behaviour characterized by the development of the vortex rope followed by a strong collapse leading to the shedding of bubbles from the runner area. Three unsteady RANS simulations are performed using the ANSYS CFX 17.2 software. The turbulence and cavitation models are, respectively, the SST and Zwart models. Regarding the time integration, a second order backward scheme is used excepted for the transport equation for the liquid volume fraction, for which a first order backward scheme is used. The simulations differ by the time step and the number of internal loops per time step. One simulation is carried out with a time step equal to one degree of revolution per time step and five internal loops. A second simulation used the same time step but 15 internal loops. The third simulations used three internal loops and an adaptive time step computed based on a maximum CFL lower than 2. The results show an influence of the time integration strategy on the cavitation volume time history both in the runner and in the draft tube with a risk of divergence of the solution if a standard set up is used.

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Kubilius, Kęstutis, and Aidas Medžiūnas. "Pathwise Convergent Approximation for the Fractional SDEs." Mathematics 10, no. 4 (February 21, 2022): 669. http://dx.doi.org/10.3390/math10040669.

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Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

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Wu, Jilian, Xinlong Feng, and Fei Liu. "Pressure-Correction Projection FEM for Time-Dependent Natural Convection Problem." Communications in Computational Physics 21, no. 4 (March 8, 2017): 1090–117. http://dx.doi.org/10.4208/cicp.oa-2016-0064.

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AbstractPressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.

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Wang, Guang. "Data Parallel Implementation of Geometric Operations." Advanced Materials Research 889-890 (February 2014): 875–80. http://dx.doi.org/10.4028/www.scientific.net/amr.889-890.875.

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The data parallel implementation scheme of zero-order interpolation and first-order interpolation of backward mapping is proved and discussed. It is shown that the complexity of data parallel implementation scheme presented in this paper is Ο(M+N) instead of Ο(MN) in sequential processing, thereby it can easily meet the real time requirements of the digital image processing.

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Britz, D., R. Baronas, E. Gaidamauskaitė, and F. Ivanauskas. "Further Comparisons of Finite Difference Schemes for Computational Modelling of Biosensors." Nonlinear Analysis: Modelling and Control 14, no. 4 (October 25, 2009): 419–33. http://dx.doi.org/10.15388/na.2009.14.4.14467.

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Simulations are presented for a reaction-diffusion system within a thin layer containing an enzyme, fed with a substrate from the surrounding electrolyte. The chemical term is of the nonlinear Michaelis-Menten type and requires a technique such as Newton iteration for solution. It is shown that approximating the nonlinear chemical term in these systems by a linearised form reduces both the accuracy and, in the case of second-order methods such as Crank-Nicolson, reduces the global error order from O(δT2) to O(δT). The first-order methods plain backwards implicit with and without linearisation, and Crank-Nicolson with linearisation are all of O(δT) and very similar in performance, requiring, for a given accuracy target, an order of magnitude more CPU time than the efficient methods backward implicit with extrapolation and Crank-Nicolson, both with Newton iteration to handle the nonlinearity. Steady state computations agree with expectations, tending to the known solutions for limiting cases. The Crank-Nicolson method shows some concentration oscillations close to the outer layer boundary but this does not propagate to the inner boundary at the electrode. The backward implicit methods do not result in such oscillations and if concentration profiles are of interest, may be preferred.

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Xiong, Hui, Liya Yao, Huachun Tan, and Wuhong Wang. "Pedestrian Walking Behavior Revealed through a Random Walk Model." Discrete Dynamics in Nature and Society 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/405907.

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This paper applies method of continuous-time random walks for pedestrian flow simulation. In the model, pedestrians can walk forward or backward and turn left or right if there is no block. Velocities of pedestrian flow moving forward or diffusing are dominated by coefficients. The waiting time preceding each jump is assumed to follow an exponential distribution. To solve the model, a second-order two-dimensional partial differential equation, a high-order compact scheme with the alternating direction implicit method, is employed. In the numerical experiments, the walking domain of the first one is two-dimensional with two entrances and one exit, and that of the second one is two-dimensional with one entrance and one exit. The flows in both scenarios are one way. Numerical results show that the model can be used for pedestrian flow simulation.

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Antil, Harbir, Ricardo H. Nochetto, and Pablo Venegas. "Optimizing the Kelvin force in a moving target subdomain." Mathematical Models and Methods in Applied Sciences 28, no. 01 (December 13, 2017): 95–130. http://dx.doi.org/10.1142/s0218202518500033.

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In order to generate a desired Kelvin (magnetic) force in a target subdomain moving along a prescribed trajectory, we propose a minimization problem with a tracking type cost functional. We use the so-called dipole approximation to realize the magnetic field, where the location and the direction of the magnetic sources are assumed to be fixed. The magnetic field intensity acts as the control and exhibits limiting pointwise constraints. We address two specific problems: the first one corresponds to a fixed final time whereas the second one deals with an unknown force to minimize the final time. We prove existence of solutions and deduce local uniqueness provided that a second-order sufficient condition is valid. We use the classical backward Euler scheme for time discretization. For both problems we prove the [Formula: see text]-weak convergence of this semi-discrete numerical scheme. This result is motivated by [Formula: see text]-convergence and does not require second-order sufficient condition. If the latter holds then we prove [Formula: see text]-strong local convergence. We report computational results to assess the performance of the numerical methods. As an application, we study the control of magnetic nanoparticles as those used in magnetic drug delivery, where the optimized Kelvin force is used to transport the drug to a desired location.

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Liu, Zhihui, and Zhonghua Qiao. "Strong approximation of monotone stochastic partial differential equations driven by white noise." IMA Journal of Numerical Analysis 40, no. 2 (January 3, 2019): 1074–93. http://dx.doi.org/10.1093/imanum/dry088.

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Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

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