Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations

England, Roland

doi:10.1093/comjnl/12.2.166

Cited by 106 publications

(32 citation statements)

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“…The Saint-Venant equations were solved numerically by the finite volume method using a central-upwind scheme [26], the fourth order Runge-Kutta method with the estimate of truncation error [27], and adaptive time step-size control [22]. The transport equations were solved by using an implicit finite differences method and applying the front limitation algorithm [28].…”

Section: Initial and Boundary Conditions Numerical Solutionmentioning

confidence: 99%

Transport of Conservative and “Smart” Tracers in a First-Order Creek: Role of Transient Storage Type

Yakirevich

Shelton

Hill

et al. 2017

Water

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Using "smart" tracers such as Resazurin (Raz) allows assessment of sediment-water interactions and associated biological activity in streams. We compared two approaches to simulate the effects of transient storage (TS) on the transport of conservative and reactive tracers. The first approach considered TS as composed of metabolically active and metabolically inactive compartments, while the second model approach accounted for the surface transient storage (STS) and hyporheic transient storage (HTS). Experimental data were collected at a perennial first-order creek in Maryland, MD, USA, by injecting the conservative tracer bromide (Br) and the reactive (Raz) tracer and sampling water at two weir stations. The STS-HTS approach led to a more accurate simulation of Br transport and tails of the Raz and its product Rezorufin (Rru) breakthrough curves. Sediments support large microbial communities, and the STS-HTS model in creeks provides additional parameters to characterize the habitats of microbial water-quality indicator organisms.

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Section: Initial and Boundary Conditions Numerical Solutionmentioning

confidence: 99%

Transport of Conservative and “Smart” Tracers in a First-Order Creek: Role of Transient Storage Type

Yakirevich

Shelton

Hill

et al. 2017

Water

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“…Steihaug and Wolfbrandt [27] examined a class of generalized Runge-Kuttas, which they called "Modified Rosenbrock" methods, and produced a (2, 2, 4, 2) method which remains consistent with an approximate Jacobian. The method uses a third order reference formula to give an error estimator of the type due to England [13]. In this method, extra stages are required to obtain the higher order reference formula.…”

Section: =1mentioning

confidence: 99%

Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Day¹,

Murthy²

1982

Math. Comp.

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Abstract. Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

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“…Hence it can model factory equipment in terms of their actual floor locations (7). ) Integration in GASP IV is done by a variable-step-size fourth-order Runge-Kutta algorithm (123,124).…”

Section: Implementation Statusmentioning

confidence: 99%

Standards for computer aided manufacturing

Evans¹,

O'Neill²,

Little³

et al. 1977

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Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations

Cited by 106 publications

References 0 publications

Transport of Conservative and “Smart” Tracers in a First-Order Creek: Role of Transient Storage Type

Transport of Conservative and “Smart” Tracers in a First-Order Creek: Role of Transient Storage Type

Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Standards for computer aided manufacturing

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