The stability of the fourth order Runge-Kutta method for the solution of systems of differential equations

Abstract

APPENDIXThere are several way.s in which one can arrange to carry out a particular calculation in various modes of arithmetic. The following particularly convenient way was suggested to us by Pi'ofes.sor W. Kahan.Most of the programming was written in FORTRAN II for the IBJM 7090 and 7094. All of the ste]:)s ia the program which were to be part of the calculation procedure under study were put on cards with a D ia column 1. As a result, these steps are compiled as double precision calculations. Double precision computations are performed in FORTRAN II by referencing a FAP subroutine which ia turn carries out the double precisioa operatioas. For exaaiple, if a double precisioa addition is to be executed, a reference is made to the entry point (DFAD) in the FAP subroutine. Similarly double precision subtraction, multiplication and division cause references to (DFSB), (DFJN'IP) and (DFDP). The arguments for these references are supplied in a standard manner.Normally, references to a subroutine will cause ihe loader to look for that subroutine ia the source deck. If the subroutine is not with the source deck the loader expects to find it in the hbrary. Thus, to provide for different modes of arithmetic oae need only .supply, along with the source deck, a modified version of the subroutine which would ordinarily be used to perform the double precision operations. Such a subroutine was wi'itten, and it was given the ability to perform any one of the five desired modes of arithmetic.In order to specify the particular mode of operation desired, one uses the FORTRAN statement CALL ARITH(I) where / = 1, 2, 3, 4 or 5 depending on which mode is required. APJTH i.s another entry point in the same subroutine. Eatry at this point causes a switch to be set which determines which mode will be used in subsequent entries to the arithmetic parts of the subroutine.The advantage of this way of implementing the various modes of calculatioa is that any calculation can easily be treated ia whatever mode oae wishes. It is necessary only to write each step of the calculation ia FORTRAN, with a D in column 1, and then to precede the whole calculation with the appropriate CALL ARITH statement. Of course the value of I can be changed for subsequent calculatioas, aad the same calculatioa can be carried out any number of times in any aiode of arithmetic.It is clear that the technique can be easily extended so that further values of I could correspond to other types of arithmetic, such as interval arithmetic or sigailicance arithmetic. The only restriction is that the speeial type of arithmetic must not use more than two memory words for each item.The above technique cannot be implemented in FOR-TRAN IV on the 7094 because double precision operations with this system use hardware iastcad of subroutines.Volume 9 / Number 2 / Febnuirv, 1%6

Application of the one-dimensional drift-flux model for numerical simulation of gas–liquid isothermal flows in vertical pipes: a mechanistic approach based on the flow pattern

“…2 on the imaginary axis, with Δz ≠ 0 [44]. The employed step size attends the RK4 stability criterion.…”

The Crossover Time as an Evaluation of Ocean Models Against Persistence

“…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.…”