C++ for Scientists - Technische Universität Dresden

14.3. THE SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS 273
The method that we use here is the Runge-Kutta 4 method, which is described here: the
solution at time step tj+1 is computed as
14.3.2 Software
Write a generic function
uj+1 = uj + h
6 (k1 + 2k2 + 2k3 + k4)
k1
k2
=
=
f(uj)
�
f uj + h
2 k1
k3 =
�
�
f uj + h
2 k2
�
k4 = f(uj + hk3)
template
void rk4( U& u, F& f, T const& h ) ;
that computes one time step with the Runge-Kutta 4 method. The argument u is on input the
solution at time t and on output at timestep t+h. The argument f is the functor that evaluates
the function f(u). The argument u is a vector.
When the implementation is finished, write the concepts for U and F in comment lines in the
code.
14.3.3 The van der Pol oscillator
Differential equations appear in the study of physical phenomena. The Van der Pol oscillator
is described by the following equation:
d2x dt2 − µ(1 − x2 ) dx
+ x = 0 (14.3)
dt
with initial solution x(0) and x ′ (0). This is a non-linear second order differential equation, with
a parameter µ. When µ = 0, we have a purely harmonic solution (cos and sin). When µ ≥ 0,
the solution evolves to a harmonic limit cycle.
Second order differential equations are usually solved by writing them as a system of first order
differential equations:
� � � � � �
d dx
dt −µ(1 − x2 dx
) 1
+
dt = 0 .
dt x
−1 0 x
In matrix form, the equation can be written as
where
A(u) =
du
dt
+ A(u)u = 0
� −µ(1 − u 2 2 ) 1
−1 0
�
,