Additive Runge-Kutta Methods for Stiff Ordinary Differential Equations

Cooper, G. J.; Sayfy, A.

doi:10.2307/2007370

Cited by 19 publications

(30 citation statements)

References 5 publications

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“…Key challenges in adapting this method to the system in Eq. (1) include finding solutions that are only periodic up to a phase; incorporating the gain, g 0 , in the adjoint system to allow other variables (such as the period) to be used as bifurcation parameters; and adapting high order, semi-implicit Runge-Kutta methods [34,35] to handle the case when the terms responsible for stiffness (those involving ∂ 2 u/∂t 2 in (1)) depend non-linearly on u through a gain g that depends on u .…”

Section: The Adjoint Continuation Methods (Acm)mentioning

confidence: 99%

“…[36], and Refs. [34,35] contain additional details about the ARK formulae. Although they are not mentioned in detail here, the theoretical aspects of the ACM have not been neglected.…”

Section: Conclusion and Experimental Verificationmentioning

confidence: 99%

“…Evolve solution forward until z = Z [34,35]. Store the solution, zderivative, and Hermite correction data [26,27] Evolve the adjoint equation until ζ = Z [34,35].…”

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

“…Store the solution, zderivative, and Hermite correction data [26,27] Evolve the adjoint equation until ζ = Z [34,35]. Use cubic Hermite interpolation to approximate X(z) [26,27] Compute G and ∇G from X(Z), X z (Z), X(Z) and X(0) [26,27] BFGS update step [38] Update initial condition as specified by BFGS [38] Has BFGS converged?…”

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

“…On the other hand, a fully implicit method would be difficult to implement due to the fact that (1) is nonlinear and (B.5), while linear, is not diagonalized by the Fourier transform. The idea of an implicit-explicit (IMEX) multi-step method [42], or an additive Runge-Kutta (ARK) method [34,35], is to write the evolution equation as a sum…”

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

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Continuation of periodic solutions in the waveguide array mode-locked laser

Williams

Wilkening

Shlizerman

et al. 2011

Physica D: Nonlinear Phenomena

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We apply the adjoint continuation method to construct highly-accurate, periodic solutions that are observed to play a critical role in the multi-pulsing transition of mode-locked laser cavities. The method allows for the construction of solution branches and the identification of their bifurcation structure. Supplementing the adjoint continuation method with a computation of the Floquet multipliers allows for explicit determination of the stability of each branch. This method reveals that, when gain is increased, the multi-pulsing transition starts with a Hopf bifurcation, followed by a period-doubling bifurcation, and a saddle-node bifurcation for limit cycles. Finally, the system exhibits chaotic dynamics and transitions to the double-pulse solutions. Although this method is applied specifically to the waveguide array mode-locking model, the multi-pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models.

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Section: The Adjoint Continuation Methods (Acm)mentioning

confidence: 99%

“…[36], and Refs. [34,35] contain additional details about the ARK formulae. Although they are not mentioned in detail here, the theoretical aspects of the ACM have not been neglected.…”

Section: Conclusion and Experimental Verificationmentioning

confidence: 99%

“…Evolve solution forward until z = Z [34,35]. Store the solution, zderivative, and Hermite correction data [26,27] Evolve the adjoint equation until ζ = Z [34,35].…”

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

See 3 more Smart Citations

Continuation of periodic solutions in the waveguide array mode-locked laser

Williams

Wilkening

Shlizerman

et al. 2011

Physica D: Nonlinear Phenomena

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Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Ambrose

Wilkening

2010

J Nonlinear Sci

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We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t = 0.We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our J Nonlinear Sci (2010) 20: 277-308 numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

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Stability of algorithms for a two domain natural convection problem and observed model uncertainty

Connors

Ganis

2010

Comput Geosci

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Two numerical algorithms are presented that couple a Boussinesq model of natural heat convection in two domains, motivated by the dynamic core of climate models. The first uses a monolithic coupling across the fluid-fluid interface. The second is a parallel implementation decoupled via a partitioned time stepping scheme with two-way communication. These new approaches are both proven to be unconditionally stable, a property not shared by traditional climate codes that use one-way coupling. Another critical property for a robust climate code is the reliability of the computation with respect to noise in unknown input parameters. This effect is measured by adding stochastic noise to two nonlinear coupling terms, involving momentum and heat transfer. Using an uncertainty quantification method known as stochastic collocation, we empirically measure the resulting variance in average surface temperature for each numerical model, including a third variant that employs one-way coupling for comparison. Throughout the duration of the simulation, we observe the smallest increase in variance using the monolithic algorithm and the largest increase using the one-way coupled algorithm.

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Additive Runge-Kutta Methods for Stiff Ordinary Differential Equations

Cited by 19 publications

References 5 publications

Continuation of periodic solutions in the waveguide array mode-locked laser

Continuation of periodic solutions in the waveguide array mode-locked laser

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

Stability of algorithms for a two domain natural convection problem and observed model uncertainty

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