scholarly journals Linear Stability of Partitioned Runge–Kutta Methods

2011 ◽  
Vol 49 (1) ◽  
pp. 232-263 ◽  
Author(s):  
R. I. McLachlan ◽  
Y. Sun ◽  
P. S. P. Tse
Keyword(s):  
Linear Stability ◽  
Runge Kutta

Linear stability of WENO schemes coupled with explicit Runge-Kutta schemes

2011 ◽  
Vol 69 (6) ◽  
pp. 1065-1095 ◽  
Author(s):  
V. Hermes ◽  
I. Klioutchnikov ◽  
H. Olivier
Keyword(s):  
Linear Stability ◽  
Weno Schemes ◽  
Runge Kutta

scholarly journals Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model

2017 ◽  
Vol 145 (9) ◽  
pp. 3817-3838 ◽  
Author(s):  
Thibaut Lunet ◽  
Christine Lac ◽  
Franck Auguste ◽  
Florian Visentin ◽  
Valéry Masson ◽  
...  
Keyword(s):  
Linear Stability ◽  
Fourth Order ◽  
Splitting Method ◽  
Time Integration ◽  
Third Order ◽  
Weno Schemes ◽  
Integration Methods ◽  
Time Integration Methods ◽  
Runge Kutta ◽  
Fifth Order

This paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme.

scholarly journals Linear Stability Analysis of Runge-Kutta Methods for Singular Lane-Emden Equations

2020 ◽  
pp. 134-140
Author(s):  
M. O. Ogunniran ◽  
O. A. Tayo ◽  
Y. Haruna ◽  
A. F. Adebisi
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Classical Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Mind

Runge-Kutta methods are efficient methods of computations in differential equations, the classical Runge-Kutta method of order 4 happens to be the most popular of these methods, and most times it is attached to the mind when Runge-Kutta methods are mentioned. However, there are numerous forms of them existing in lower and higher orders of the classical method. This work investigates the linear stabilities and abilities of some selected explicit members of these Runge-Kutta methods in integrating the singular Lane-Emden differential equations. The results obtained established the ability of the classical Runge-Kutta method and why is mostly used in computations.

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