Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems

Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave–slow-wave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on Adams methods or on backward differencing. Two new families of methods are proposed that have good stability properties in fast-wave–slow-wave problems: one family is based on Adams methods and the other on backward schemes. Here the focus is primarily on four specific schemes drawn from these two families: a pair of Adams methods and a pair of backward methods that are either (i) optimized for third-order accuracy in the explicit component of the full IMEX scheme, or (ii) employ particularly good schemes for the implicit component. These new schemes are superior, in many respects, to the linear multistep IMEX schemes currently in use. The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in fully nonlinear simulations of gravity waves generated by a localized source in a shear flow.

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IMEX peer methods for fast-wave–slow-wave problems

Applied Numerical Mathematics ◽

10.1016/j.apnum.2017.02.016 ◽

2017 ◽

Vol 118 ◽

pp. 221-237 ◽

Cited By ~ 6

Author(s):

Behnam Soleimani ◽

Oswald Knoth ◽

Rüdiger Weiner

Keyword(s):

Slow Wave ◽

Fast Wave ◽

Wave Problems

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Implicit–Explicit Runge–Kutta Methods for Fast–Slow Wave Problems

Monthly Weather Review ◽

10.1175/mwr-d-13-00132.1 ◽

2013 ◽

Vol 141 (10) ◽

pp. 3426-3434 ◽

Cited By ~ 5

Author(s):

Jeffrey S. Whitaker ◽

Sajal K. Kar

Keyword(s):

Slow Wave ◽

Time Integration ◽

Forecast Model ◽

Second Order ◽

Water Model ◽

Fast Wave ◽

Integration Schemes ◽

Time Integration Schemes ◽

Runge Kutta ◽

Wave Problems

Abstract Linear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need not be accurately simulated are investigated. The authors focus on three-stage, second-order schemes and show that a scheme recently proposed by one of them (Kar) is unstable for purely oscillatory problems. The instability is reduced if the averaging inherent in the implicit part of the scheme is decentered, sacrificing second-order accuracy. Two alternative schemes are proposed with better stability properties for purely oscillatory problems. One of these utilizes a 3-cycle Lorenz scheme for the slow-wave terms and a trapezoidal scheme for the fast-wave terms. The other is a combination of two previously proposed schemes, which is stable for purely oscillatory problems for all fast-wave frequencies when the slow-wave frequency is less than a critical value. The alternative schemes are tested using a global spectral shallow-water model and a version of the NCEP operational global forecast model. The accuracy and stability of the alternative schemes are discussed, along with their computational efficiency.

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Stability analysis of linear multistep methods for delay differential equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171286000583 ◽

1986 ◽

Vol 9 (3) ◽

pp. 447-458 ◽

Cited By ~ 6

Author(s):

V. L. Bakke ◽

Z. Jackiewicz

Keyword(s):

Stability Analysis ◽

Differential Equations ◽

Delay Differential Equations ◽

Lagrange Interpolation ◽

Multistep Methods ◽

Linear Multistep Methods ◽

Stability Properties ◽

Adams Methods ◽

Test Equation ◽

Delay Differential

Stability properties of linear multistep methods for delay differential equations with respect to the test equationy′(t)=ay(λt)+by(t), t≥0,0<λ<1, are investigated. It is known that the solution of this equation is bounded if and only if|a|<−band we examine whether this property is inherited by multistep methods with Lagrange interpolation and by parametrized Adams methods.

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High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-020-00110-5 ◽

2021 ◽

Author(s):

Giacomo Albi ◽

Lorenzo Pareschi

Keyword(s):

Multistep Methods ◽

General Setting ◽

Reaction Diffusion ◽

Strong Stability ◽

High Order ◽

Iterative Solvers ◽

Time Dependent ◽

Linear Multistep Methods ◽

Advection Diffusion Equation ◽

Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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