2.2 Computational Tools
35
Algorithmic Time Integration
For a generic (autonomous) evolution problem the corresponding first-order evolution equation in time for the evolving variable x = x(t) reads together with appropriate initial conditions at time t = 0 as
˙
x(t) = f
x(t)
for t ∈ T = [0, T ] with x(0) = x
0
.
(2.31)
Then, in principle, the solution trajectory x(t) within the time interval of interest
T = [0, T ] follows analytically from integrating the evolution equation
x(t) − x
0
=
t
0
f
x(t)
dt for t ∈ [0, T ].
(2.32)
In many cases analytical integration of the evolution equation is however not possible
or at least cumbersome. Thus approximations x
n to the discrete instants x(t
n
) of the
solution trajectory x(t) are sought, i.e.
x
n
≈ x(t
n
) for n ∈ [0, n mx ].
(2.33)
Then the approximations x
n to the discrete instants x(t
n
) of the solution trajectory
x(t) are typically obtained by a one-step (and one-stage) integration algorithm, i.e. an
integration algorithm that involves only the discrete solutions x
n−1 and x
n at the start
and end of one time step. Such an integration algorithm is generically represented in
terms of the time step size t
n and an integration rule
x
n
− x
n−1
= t
n RULE{ f
x(t)
, x
n
, x
n−1
}.
(2.34)
Thereby the integration rule depends on the format f
x(t)
of the evolution equation, possibly the unknown approximation x
n to x(t
n
), and also possibly the known
approximation x
n−1 to x(t
n−1
).
Integration algorithms depending only on the known approximation x
n−1 to
x(t
n−1
) are denoted as explicit, whereas integration algorithms depending in one
way or other on the unknown approximation x
n to x(t
n
) are termed implicit. Further
important criteria to discriminate between different integration algorithms are their
conditional or unconditional stability and their order of accuracy.
Typical representatives of one-step integration algorithms as showcased in Fig. 2.7
are for example:
The Euler Forward Integrator
x
n
− x
n−1
= t
n f (x
n−1
).
(2.35)
Explicit, conditionally stable, and first-order accurate.
35
Algorithmic Time Integration
For a generic (autonomous) evolution problem the corresponding first-order evolution equation in time for the evolving variable x = x(t) reads together with appropriate initial conditions at time t = 0 as
˙
x(t) = f
x(t)
for t ∈ T = [0, T ] with x(0) = x
0
.
(2.31)
Then, in principle, the solution trajectory x(t) within the time interval of interest
T = [0, T ] follows analytically from integrating the evolution equation
x(t) − x
0
=
t
0
f
x(t)
dt for t ∈ [0, T ].
(2.32)
In many cases analytical integration of the evolution equation is however not possible
or at least cumbersome. Thus approximations x
n to the discrete instants x(t
n
) of the
solution trajectory x(t) are sought, i.e.
x
n
≈ x(t
n
) for n ∈ [0, n mx ].
(2.33)
Then the approximations x
n to the discrete instants x(t
n
) of the solution trajectory
x(t) are typically obtained by a one-step (and one-stage) integration algorithm, i.e. an
integration algorithm that involves only the discrete solutions x
n−1 and x
n at the start
and end of one time step. Such an integration algorithm is generically represented in
terms of the time step size t
n and an integration rule
x
n
− x
n−1
= t
n RULE{ f
x(t)
, x
n
, x
n−1
}.
(2.34)
Thereby the integration rule depends on the format f
x(t)
of the evolution equation, possibly the unknown approximation x
n to x(t
n
), and also possibly the known
approximation x
n−1 to x(t
n−1
).
Integration algorithms depending only on the known approximation x
n−1 to
x(t
n−1
) are denoted as explicit, whereas integration algorithms depending in one
way or other on the unknown approximation x
n to x(t
n
) are termed implicit. Further
important criteria to discriminate between different integration algorithms are their
conditional or unconditional stability and their order of accuracy.
Typical representatives of one-step integration algorithms as showcased in Fig. 2.7
are for example:
The Euler Forward Integrator
x
n
− x
n−1
= t
n f (x
n−1
).
(2.35)
Explicit, conditionally stable, and first-order accurate.