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Differential equations may be specified as explicit systems of first order.
Example: Horizontal Throw

CONSTANTS
m  (REAL [kg])    := 0.2  [kg],
g  (REAL [m/s^2]) := 9.81 [m/s^2],
vx (REAL [m/s])   := 20   [m/s],

STATE VARIABLES
CONTINUOUS
y  (REAL [m])   := 2.0 [m],
vy (REAL [m/s]) := 0.0 [m/s]

DIFFERENTIAL EQUATIONS
y'  := vy;
vy' := -m*g;
END

Implicit systems can be described by iteratively formulated algebraic equations .

Partial differential equations are handled by spatial discretisation .

Differential equations with discontinuous differential quotients and the combination with discrete events attract special attention.

The following methods of integration may be selected (order of error in parantheses):

Explicit Methods:
  • Euler (1)
  • improved Euler (2)
  • Heun (2)
  • classical Runge-Kutta (4)
  • Runge-Kutta-England (4/5)
  • Runge-Kutta-Fehlberg (5/6)
Implicit Methods:
  • implicit Euler method (1)
  • implicit Heun-method (2)
  • implicit Runge-Kutta-method of Gauß-type (6)
  • Rosenbrock-Wanner (4)
Special Methods:
  • Multistep method of Adams-Bashforth-Moulton (6/7)
  • Extrapolation method of Bulirsch-Stör (order adaptive)

The step size is controlled by the relative error of integration for which the user can specify an upper limit.

The error estimation is done through

  • bisection of step size
  • doubling of step size
  • increase of error order.