Economical evaluation of Runge-Kutta formulae

Abstract

1. Gill [1] and Blum [2] have produced special versions of the Runge-Kutta fourth order method for the solution of N simultaneous first order differential equations which require 3A 4-P storage locations against the normal 4JV -f P, where P is the storage required by the program. It is shown below that it is possible to arrange all such methods in a form which requires SN -\-P storage locations. Gill's method for reducing round off error is also extended. a4-&4-c4-d= 1, 6m 4-en 4-dp = |, bm 4-en 4-dp" = \, cmr 4-dnt -\-dms = -g-, The computation with the formulae arranged in this form requires 4A 4-P storage

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

“…The classical, four-stage, fourth-order, low-storage form of the Runge-Kutta method (RK4) [18] and the second-order fully implicit scheme (Imp2), called ADI-SGS scheme [19,20], are used for time integration. The ADI-SGS scheme is derived by combining alternative direction implicit (ADI) factorization [21] with the lower-upper symmetric-Gauss-Seidel (LU-SGS) method [22].…”

High-order compact finite-difference methods on general overset grids

“…The explicit approach employs the classical fourth-order Runge-Kutta method (RK4) implemented in lowstorage form [35], and is utilized for wave propagation problems. Problems requiring very fine resolution such as wall-bounded viscous flow are handled with the approximately factored Beam-Warming (BW) implicit method [36].…”

Runge–Kutta methods with minimum storage implementations

“…Throughout the development of low-storage Runge-Kutta methods, fourth-order methods have been of particular interest [10,2,9,21]. In Table 1 we present several examples of minimum storage fourth-order methods.…”

“…Blum later provided a three-register implementation of the classical Runge-Kutta method (with rational coefficients) [2]. Fyfe showed that all four-stage fourth-order methods are capable of three-register implementation [9]. Shampine devised a variety of techniques for reducing the storage requirements of methods with many stages [16].…”