NUMAL Section 5.2.1.1.1.1

BEGIN SECTION : 5.2.1.1.1.1 (December, 1979)

SECTION 5.2.1.1.1.1 CONTAINS NINE PROCEDURES FOR INTIAL-VALUE PROBLEMS
FOR FIRST ORDER  ORDINARY DIFERENTIAL EQUATIONS.

Section 5.2.1.1.1.1.A:
A.  RK1 SOLVES AN IVP FOR A SINGLE ODE BY MEANS OF A
    5-TH ORDER RUNGE-KUTTA METHOD.

Section 5.2.1.1.1.1.B:
B.  RKE SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A
    5-TH ORDER RUNGE-KUTTA METHOD.

Section 5.2.1.1.1.1.C:
C.  RK4A SOLVES AN IVP FOR A SINGLE ODE BY MEANS OF A 5-TH ORDER
    RUNGE-KUTTA METHOD. RK4A INTERCHANGES THE DEPENDENT AND
    INDEPENDENT VARIABLE.

Section 5.2.1.1.1.1.D:
D.  RK4NA SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A
    5-TH ORDER RUNGE-KUTTA METHOD. RK4NA INTERCHANGES THE
    INDEPENDENT AND ONE DEPENDENT VARIABLE.

Section 5.2.1.1.1.1.E:
E.  RK5NA SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A
    5-TH ORDER RUNGE-KUTTA METHOD. RK5NA USES THE ARC LENGTH
    AS INTEGRATION VARIABLE.

Section 5.2.1.1.1.1.F:
F.  MULTISTEP  SOLVES  AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A
    LINEAR MULTISTEP METHOD. IT USES EITHER THE BACKWARD
    DIFFERENTIATION METHODS, OR THE ADAMS-BASHFORTH-MOULTON-METHOD.

Section 5.2.1.1.1.1.G:
G.  DIFFSYS  SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF
    A HIGH ORDER EXTRAPOLATION METHOD BASED ON THE MODIFIED
    MIDPOINT RULE.

Section 5.2.1.1.1.1.H:
H.  ARK SOLVES AN IVP FOR A LARGE SYSTEM OF ODE'S WHICH IS OBTAINED
    FROM SEMI-DISCRETIZATION OF AN INITIAL BOUNDARY VALUE PROBLEM
    FOR A PARABOLIC OR HYPERBOLIC EQUATION. ARK IS BASED ON
    STABILIZED, EXPLICIT RUNGE-KUTTA METHODS OF LOW ORDER.

Section 5.2.1.1.1.1.I:
I.  EFRK  SOLVES  AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF AN
    EXPONENTIALLY FITTED EXPLICIT RUNGE-KUTTA METHOD OF FIRST,
    SECOND OR THIRD ORDER.

RK1 AND RKE ARE INTENDED FOR NON-STIFF EQUATIONS,WHILE RK4A, RK4NA
AND RK5NA   ARE INTENDED FOR NON-STIFF EQUATIONS WHERE DERIVATIVES
BECOME VERY LARGE, SUCH AS IN THE NEIGHBOURHOOD OF SINGULARITIES.
MULTISTEP CAN BE USED FOR NON-STIFF, AS WELL AS STIFF EQUATIONS.
DIFFSYS SHOULD BE USED FOR PROBLEMS FOR WHICH A VERY HIGH ACCURACY
IS DESIRED. ARK IS RECOMMENDED FOR THE INTEGRATION OF SEMI-DISCRETE
PARABOLIC OR HYPERBOLIC PROBLEMS. EFRK IS A SPECIAL PURPOSE PROCEDURE
FOR STIFF EQUATIONS WITH A KNOWN, CLUSTERED EIGENVALUE SPECTRUM.
EXCEPT EFRK, ALL PROCEDURES PERFORM STEPSIZE CONTROL.