Stability of the Fourth Order Runge-Kutta Method for the Solution of Systems of Differential Equations

Karim, Abbas I. Abdel

doi:10.1093/comjnl/9.3.308

Cited by 5 publications

(2 citation statements)

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“…To compute the separation distance (equation (2)), the trajectories for both the persistence and MERCATOR models are advected using a fourth-order Runge-Kutta advection scheme (Abdel Karim, 1966). For simplicity, only the advection (i.e., the forcing of the ocean currents alone) acts on the simulated trajectories.…”

Section: Methodsmentioning

confidence: 99%

The Crossover Time as an Evaluation of Ocean Models Against Persistence

Phillipson

Toumi

2018

Geophysical Research Letters

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A new ocean evaluation metric, the crossover time, is defined as the time it takes for a numerical model to equal the performance of persistence. As an example, the average crossover time calculated using the Lagrangian separation distance (the distance between simulated trajectories and observed drifters) for the global MERCATOR ocean model analysis is found to be about 6 days. Conversely, the model forecast has an average crossover time longer than 6 days, suggesting limited skill in Lagrangian predictability by the current generation of global ocean models. The crossover time of the velocity error is less than 3 days, which is similar to the average decorrelation time of the observed drifters. The crossover time is a useful measure to quantify future ocean model improvements.

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Section: Methodsmentioning

confidence: 99%

The Crossover Time as an Evaluation of Ocean Models Against Persistence

Phillipson

Toumi

2018

Geophysical Research Letters

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“…Thus, the scheme (13) is stable. For a detailed mathematical proof of the stability of the Runge-Kutta method, the reader may read the references [31,32] and therein.…”

Section: Stability Analysismentioning

confidence: 99%

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Kutluay

Karta

Yağmurlu

2019

Numerical Methods Partial

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In this article, the generalized Rosenau-KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie-Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge-Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L 2 and L ∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.

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