Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

Layton, William; Pei, Wenlong; Qin, Yi; Trenchea, Catalin

doi:10.1002/num.22831

Cited by 12 publications

(12 citation statements)

References 39 publications

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“…Its detailed specification (given in Section 2), is sufficiently Gordian to deter its use in complex applications, in which a method with DLN's excellent properties should be valued. Our preliminary work on adaptive time-stepping for flow problems [4,5] shows that (DLN) has promise, motivating the work herein.…”

Section: Introductionmentioning

confidence: 91%

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Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Layton¹,

Pei²,

Trenchea³

2021

Preprint

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The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional G-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method is equivalent to the backward Euler method with pre-and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.

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Section: Introductionmentioning

confidence: 91%

“…The 'energy' identity (3.2), implicit in [2], follows from algebraic manipulations, see e.g. [4]. The G-stability of (DLN), i.e.…”

mentioning

confidence: 99%

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Layton¹,

Pei²,

Trenchea³

2021

Preprint

Self Cite

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“…The simple tests of the variable step DLN method in [6,8,52,55] have confirmed its potential. The work herein is complemented by further DLN analysis applied to fluids problems in [57,70], and refactorization to reduce the cognitive complexity of implementation in [42,58].…”

Section: Related Workmentioning

confidence: 99%

Time step adaptivity in the method of Dahlquist, Liniger and Nevanlinna

Layton

Pei

Trenchea

2023

Preprint

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Dahlquist, Liniger and Nevanlinna devised a family of one-leg two-step methods (DLN) that is second order, A- and G- stable for arbitrary, non-uniform time steps. The DLN method thus has strong potential for use in adaptive codes, but its adaptive step size selection is little explored. This report develops two approaches for the efficient local error estimation in the DLN method, and tests their use in a standard adaptivity framework. Many methods of error estimation are possible; herein we focus on two complementary estimators which involve minimal extra storage and computations. First we evaluate the local truncation error of the DLN method by Milne’s device, using the difference between the solution of the DLN method and the solution of a variable-step, explicit, second-order Adams-Bashforth-like method. Second, we use a recent refactorization of the DLN method, which eases implementation of DLN in legacy codes, to obtain an effective error estimation at no extra cost. We perform a number of numerical tests, comparing the two time adaptive DLN algorithms with some standard numerical ODE packages. Our tests indicate that the adaptive DLN method, with error estimated by Milne’s device, is an efficient and reliable method, even for stiff and unstable problems.

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“…It was shown under a timestep condition that both methods were long time energy stable. Recently, for the NSE adaptive time stepping schemes have been studied for a variety of second order implicit and linearly implicit methods [15,17,18,5]. It was demonstrated that time adaptivity increased the accuracy and efficiency of the schemes.…”

Section: Previous Workmentioning

confidence: 99%

“…For the last nonlinear term in (18), adding and subtracting b * (E n+1 (u h ), u n+1 h , ϕ n+1 h ), and using the skew-symmetry of the nonlinear term yields…”

Section: Error Analysismentioning

confidence: 99%

An embedded variable step IMEX scheme for the incompressible Navier-Stokes equations

DeCaria,

Schneier

2020

Preprint

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This report presents a series of implicit-explicit (IMEX) variable timestep algorithms for the incompressible Navier-Stokes equations (NSE). With the advent of new computer architectures there has been growing demand for low memory solvers of this type. The addition of time adaptivity improves the accuracy and greatly enhances the efficiency of the algorithm. We prove energy stability of an embedded first-second order IMEX pair. For the first order member of the pair, we prove stability for variable stepsizes, and analyze convergence. We believe this to be the first proof of this type for a variable step IMEX scheme for the incompressible NSE. We then define and test a variable stepsize, variable order IMEX scheme using these methods. Our work contributes several firsts for IMEX NSE schemes, including an energy argument and error analysis of a two-step, variable stepsize method, and embedded error estimation for an IMEX multi-step method. Variable Step BE-AB2 Scheme; Implicit/Explicit; IMEX; Navier-Stokes; Time Filters

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Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

Cited by 12 publications

References 39 publications

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Time step adaptivity in the method of Dahlquist, Liniger and Nevanlinna

An embedded variable step IMEX scheme for the incompressible Navier-Stokes equations

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