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An implicit–explicit second order BDF numerical scheme with variable steps for gradient flows [article]

Dianming Hou, Zhonghua Qiao
2022 arXiv   pre-print
In this paper, we propose and analyze an efficient implicitexplicit (IMEX) second order in time backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using  ...  We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio γ_n+1:=_n+1/_n≤ 4.8645.  ...  In this paper, we propose and analyze an efficient implicit-explicit BDF2 scheme with variable time steps for gradient flow problems using the SAV approach.  ... 
arXiv:2204.00233v1 fatcat:igklm37i4jagjajfan63ukmi2u

A comparative study of fractional step method in its quasi-implicit, semi-implicit and fully-explicit forms for incompressible flows

Rhodri LT Bevan, Etienne Boileau, Raoul van Loon, R.W. Lewis, P Nithiarasu
2016 International journal of numerical methods for heat & fluid flow  
Since the third order BDF is not 'self-starting', an adaptive procedure is implemented, with a first order BDF for the first time step, and a second order BDF for the second time step.  ...  For the second order schemes, an equivalent velocity error is achieved with even greater efficiency. This is also true when compared to the third order fully-explicit scheme.  ... 
doi:10.1108/hff-06-2015-0233 fatcat:77r342jj3ze4thli4zt5b6wqee

Nonlinear operator integration factor splitting for the shallow water equations

Amik St-Cyr, Stephen J. Thomas
2005 Applied Numerical Mathematics  
The new scheme is implemented in a spectral element shallow water model using an implicit second order backward differentiation formula for Coriolis and gravity wave terms.  ...  An analysis of splitting errors reveals that OIFS is compatible with the order conditions for linear multi-step methods.  ...  Acknowledgements The authors would like to thank Ram Nair for reviewing the manuscript and providing many helpful suggestions.  ... 
doi:10.1016/j.apnum.2004.08.038 fatcat:vzedkmg3q5bfjglkshbg36gydi

Comparing Numerical Solution Methods for the Cahn-Hilliard Equation

Riley Fisher
2021 SIAM Undergraduate Research Online  
The preferred high-order solver was an implicit-explicit Runge-Kutta scheme, which had a relatively low computational cost.  ...  The Cahn-Hilliard equation, which is derived from a gradient flow of an energy functional, is a higher-order, parabolic nonlinear partial differential equation.  ...  In particular, the authors recognize Professors Lenhard Ng and Heekyoung Hahn for their involvement in organizing DOmath 2020.  ... 
doi:10.1137/20s1357974 fatcat:6tri6gfjnfhlfeceakv4gdqo5i

Energy-stable backward differentiation formula type fourier collocation spectral schemes for the Cahn-Hilliard equation

Jun Zhou, Ke-Long Cheng
2022 Thermal Science  
We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with a Fourier collocation spectral approximation in space.  ...  An addition-al Douglas-Dupont regularization term is introduced, which ensures the energy stability with a mild requirement.  ...  We compare the numerical simulation result with the predicted coarsening rate, using the proposed second order scheme BDF-2 in (8) for the CH flow (2).  ... 
doi:10.2298/tsci2202095z fatcat:gn3cthm4ljd37hiyius5eo7m6u

Nonlinear OIFS for a Hybrid Galerkin Atmospheric Model [chapter]

St. -Cyr Amik, Stephen J. Thomas
2005 Lecture Notes in Computer Science  
An implicit second order backward differentiation formula is applied to Coriolis and gravity wave terms. The implicit system is then discretised using a continuous Galerkin spectral element method.  ...  The advantages of such an approach include improved mass and energy conservation properties. A strong-stability preserving Runge-Kutta scheme is applied for substepping.  ...  For a second order BDF-2 scheme, sub-stepping of the rhs terms is not required because Q t n S (t n ) = I.  ... 
doi:10.1007/11428862_8 fatcat:qglzq2ocfrdypegvc5nv34ib3a

Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs [article]

Justin Baker and Hedi Xia and Yiwei Wang and Elena Cherkaev and Akil Narayan and Long Chen and Jack Xin and Andrea L. Bertozzi and Stanley J. Osher and Bao Wang
2022 arXiv   pre-print
These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and accuracy guarantees.  ...  The proximal implicit ODE solver guarantees superiority over explicit solvers in numerical stability and computational efficiency.  ...  Each iteration of higher-order BDF schemes requires more NFEs than lower-order BDF schemes, showing a tradeoff between controlling numerical error and using high-order schemes. (15) .  ... 
arXiv:2204.08621v1 fatcat:oig6u3uwsjfsvpdm7szklyzn44

Automatic Differentiation using Operator Overloading (ADOO) for implicit resolution of hyperbolic single phase and two-phase flow models [article]

François Fraysse
2019 arXiv   pre-print
Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart.  ...  Implicit time integration schemes are widely used in computational fluid dynamics numerical codes to speed-up computations.  ...  Implicit schemes use single-step first order BDF and second order Newton-SSPSDIRK both using CFL equal to 20.  ... 
arXiv:1904.02136v1 fatcat:tah4brblyjhbvps7qgyktwid2y

A new class of efficient and robust energy stable schemes for gradient flows [article]

Jie Shen, Jie Xu, Jiang Yang
2017 arXiv   pre-print
By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a large class of gradient flows.  ...  We propose a new numerical technique to deal with nonlinear terms in gradient flows.  ...  We describe below how to construct higher order schemes for gradient flows by combining the SAV approach with higher order BDF schemes, and how to implement adaptive time stepping to further increase the  ... 
arXiv:1710.01331v1 fatcat:btpse2krnrfkzhmlpqpsx2u4za

Higher-order implicit-explicit multi-domain compressible Navier-Stokes solvers

Oscar P. Bruno, Max Cubillos, Edwin Jimenez
2019 Journal of Computational Physics  
., Higher-order implicit-explicit multi-domain compressible Navier-Stokes solvers, J. Comput. Phys. (2019), https://doi.  ...  We remark that for numerical stability, a mild Padé-type sixth-order filter [38] is used with both explicit and implicit time marching.  ...  For example, a processor assigned to an implicit zone can proceed with the first subiteration in parallel with the explicit solves.  ... 
doi:10.1016/ fatcat:pf5bi6qtdfd4dgsd2cwa3fwcpu

A comparative study of implicit Jacobian-free Rosenbrock-Wanner, ESDIRK and BDF methods for unsteady flow simulation with high-order flux reconstruction formulations [article]

Lai Wang, Meilin Yu
2019 arXiv   pre-print
We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order  ...  In general, ESDIRK methods allow a larger physical time step size for unsteady flow simulation than ROW methods do.  ...  DMS-0821311), with additional substantial support from UMBC.  ... 
arXiv:1904.04825v1 fatcat:jyksi7lo7zefdcrav3ohwcbhzi

Fully Implicit Spectral Boundary Integral Computation of Red Blood Cell Flow [article]

Pei Chuan Chao, Ali Gürbüz, Frederick Sachs, M. V. Sivaselvan
2021 arXiv   pre-print
The IODE is integrated in time using a multi-step implicit method based on backward difference formulas, with variable order and adaptive time-stepping controlled by local truncation error and convergence  ...  An approach is presented for implicit time integration in computations of red blood cell flow by a spectral boundary integral method.  ...  In the following, various numerical performance aspects of the implicit time integration scheme are discussed: (i) Time step size: The time-step sizes used by the implicit scheme and the explicit scheme  ... 
arXiv:2104.10214v1 fatcat:xfhdjkjkdrf7tpg3fql3vsrguq

Higher-order in time "quasi-unconditionally stable" ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

Oscar P. Bruno, Max Cubillos
2016 Journal of Computational Physics  
The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain  ...  Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based  ...  MC also thanks the National Physical Science Consortium for their support of this effort.  ... 
doi:10.1016/ fatcat:bspf3bxxgvcjljq4phz5lwsosa

A Relaxation Scheme for the Simulation of Low Mach Number Flows [chapter]

Emanuela Abbate, Angelo Iollo, Gabriella Puppo
2017 Springer Proceedings in Mathematics & Statistics  
For a second order implicit approximation, a BDF (Backward Differentiation Formula) second order accurate is employed.  ...  Nozzle flow: pressure profiles obtained with first and second order implicit schemes and with first order explicit-upwind relaxation scheme of [5] (500 grid points).  ... 
doi:10.1007/978-3-319-57394-6_25 fatcat:quhjhf2eiba7hovsklvwpoki3u

Numerical Methods for Reacting Gas Flow Simulations [chapter]

S. van Veldhuizen, C. Vuik, C. R. Kleijn
2006 Lecture Notes in Computer Science  
In this study various numerical schemes for simulating 2D laminar reacting gas flows, as typically found in Chemical Vapor Deposition (CVD) reactors, are proposed and compared.  ...  Model for CVD Simulation The mathematical model describing the CVD process consists of a set of PDEs with appropriate boundary and initial conditions, which describe the gas flow, the transport of energy  ...  The Newton iteration is, with its second order convergence, an obvious choice.  ... 
doi:10.1007/11758525_2 fatcat:4echuiprk5bi5j7ruamyyn4jzq
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