A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations

Byrne, George D.; Hindmarsh, A.C.

doi:10.1145/355626.355636

Cited by 294 publications

(167 citation statements)

References 15 publications

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“…Two di¡erent cases are considered : X Ã 6 1 when the critical concentration of crystals is reached in the conduit, and X Ã s 1 when the critical concentration of crystals is not reached and thus the viscosity in the conduit is approximated as constant. The equations are solved by means of a variable-step size, variable-order backward di¡erentiation method, suitable for sti¡ problems [15]. To compute V(d3X Ã ) the velocity is stored in an array and approximated by means of a third-order Legendre polynomial using the nearest points.…”

Section: Model Descriptionmentioning

confidence: 99%

Periodic behavior in lava dome eruptions

Бармин

Melnik

Sparks

2002

Earth and Planetary Science Letters

150

136

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Lava dome eruptions commonly display fairly regular alternations between periods of high activity and periods of low or no activity. The time scale for these alternations is typically months to several years. Here we develop a generic model of magma discharge through a conduit from an open-system magma chamber with continuous replenishment. The model takes account of the principal controls on flow, namely the replenishment rate, magma chamber size, elastic deformation of the chamber walls, conduit resistance, and variations of magma viscosity, which are controlled by degassing during ascent and kinetics of crystallization. The analysis indicates a rich diversity of behavior with periodic patterns similar to those observed. Magma chamber size can be estimated from the period with longer periods implying larger chambers. Many features observed in volcanic eruptions such as alternations between periodic behaviors and continuous discharge, sharp changes in discharge rate, and transitions from effusive to catastrophic explosive eruption can be understood in terms of the non-linear dynamics of conduit flows from open-system magma chambers. The dynamics of lava dome growth at Mount St. Helens (1980^1987) and Santiaguito (1922^2000) was analyzed with the help of the model. The best-fit models give magma chamber volumes of V0.6 km 3 for Mount St. Helens and V65 km 3 for Santiaguito. The larger magma chamber volume is the major factor in explaining why Santiaguito is a long-lived eruption with a longer periodicity of pulsations in comparison with Mount St. Helens. ß

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Section: Model Descriptionmentioning

confidence: 99%

Periodic behavior in lava dome eruptions

Бармин

Melnik

Sparks

2002

Earth and Planetary Science Letters

150

136

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“…Suffice it to say, it is derived by considering additional polynomial fits to the solution y(t). 50,51 The solution gives the desired corrected integrated population, y(t m )ϭu.…”

Section: A Direct Time Propagation With a Stiff Ode Integratormentioning

confidence: 99%

Fast, scalable master equation solution algorithms. III. Direct time propagation accelerated by a diffusion approximation preconditioned iterative solver

Frankcombe

Smith

2003

The Journal of Chemical Physics

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Articles you may be interested inIn this paper we propose a novel fast and linearly scalable method for solving master equations arising in the context of gas-phase reactive systems, based on an existent stiff ordinary differential equation integrator. The required solution of a linear system involving the Jacobian matrix is achieved using the GMRES iteration preconditioned using the diffusion approximation to the master equation. In this way we avoid the cubic scaling of traditional master equation solution methods and maintain the low temperature robustness of numerical integration. The method is tested using a master equation modelling the formation of propargyl from the reaction of singlet methylene with acetylene, proceeding through long lived isomerizing intermediates.

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“…An example of an algorithm based on this approach is CHEMEQ, which has been proposed by Young and Boris [15,16] and recommended in [II,13]. Early comparisons with the implicit backward differentiation (Gear type) codes DIFSUB [15,6] and EPISODE [11,2] are in favour of the PSSA approach and also in more recent work this approach is still applied and advocated [8,9,17,18].…”

mentioning

confidence: 99%

An Evaluation of Explicit Pseudo-Steady-State Approximation Schemes for Stiff ODE Systems from Chemical Kinetics

1994

Journal of Computational Physics

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A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations

Cited by 294 publications

References 15 publications

Periodic behavior in lava dome eruptions

Periodic behavior in lava dome eruptions

Fast, scalable master equation solution algorithms. III. Direct time propagation accelerated by a diffusion approximation preconditioned iterative solver

An Evaluation of Explicit Pseudo-Steady-State Approximation Schemes for Stiff ODE Systems from Chemical Kinetics

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