7.2 The Semi-implicit Method
Then the
dispersion relation (7.22) may be expressed as
W = _1_
and the phase speed of the
solution becomes
345
W
CI = k =
The preceding differ from the corresponding expressions for the leapfrog scheme
(7.20) and (7.21) in that the true propagation speed, c, has been replaced by the
reduced speed c. As the time step increases, c decreases, so that
remains
less than one and the numerical solution remains stable, but the relative error in c
can become arbitrarily large . As a consequence, it is not possible to take advantage of the unconditional stability of the trapezoidal method by using very large
time steps to solve wave-propagation problems unless one is willing to tolerate a
considerable decrease in the accuracy of the solution.
7.2.2 A Prototype Problem
The loss of accuracy associated with the poor temporal resolution that can occur
using implicit numerical methods is not a problem if the poorly resolved waves are
not physically significant. If the fastest-moving waves are not physically significant, the accuracy constraints imposed on the time step by these waves can be ignored, and provided that the scheme is unconditionally stable, a good solution can
be obtained using any time step that adequately resolves the slower-moving features of primary physical interest. A simple but computationally inefficient way
to ensure the unconditional stability of a numerical scheme is to use trapezoidal
time-differencing throughout the approximate equations. It is, however, more efficient to implicitly evaluate only those terms in the goveming equations that are
crucial to the propagation of the fast wave and to approximate the remaining terms
with some explicit time-integration scheme. This is the fundamental strategy in
the "serni-implicit" approach, which gains efficiency relative to a "fully implicit"
method by reducing the complexity of the implicit algebraic equations that must
be solved during each integration step. Semi-implicit differencing is particularly
attractive when all the terms that are evaluated implicitly are linear functions of
the unknown variables.
In order to investigate the stability of semi-implicit time-differencing schemes,
consider a prototype ordinary differential equation of the form
(7.23)
This is simply aversion of the oscillation equation (2.30) in which the oscillatory
forcing is divided into high -frequency (WH) and low -frequency (WL) components.
The division of the forcing into two terms may appear to be rather artificial, but the
Then the
dispersion relation (7.22) may be expressed as
W = _1_
and the phase speed of the
solution becomes
345
W
CI = k =
The preceding differ from the corresponding expressions for the leapfrog scheme
(7.20) and (7.21) in that the true propagation speed, c, has been replaced by the
reduced speed c. As the time step increases, c decreases, so that
remains
less than one and the numerical solution remains stable, but the relative error in c
can become arbitrarily large . As a consequence, it is not possible to take advantage of the unconditional stability of the trapezoidal method by using very large
time steps to solve wave-propagation problems unless one is willing to tolerate a
considerable decrease in the accuracy of the solution.
7.2.2 A Prototype Problem
The loss of accuracy associated with the poor temporal resolution that can occur
using implicit numerical methods is not a problem if the poorly resolved waves are
not physically significant. If the fastest-moving waves are not physically significant, the accuracy constraints imposed on the time step by these waves can be ignored, and provided that the scheme is unconditionally stable, a good solution can
be obtained using any time step that adequately resolves the slower-moving features of primary physical interest. A simple but computationally inefficient way
to ensure the unconditional stability of a numerical scheme is to use trapezoidal
time-differencing throughout the approximate equations. It is, however, more efficient to implicitly evaluate only those terms in the goveming equations that are
crucial to the propagation of the fast wave and to approximate the remaining terms
with some explicit time-integration scheme. This is the fundamental strategy in
the "serni-implicit" approach, which gains efficiency relative to a "fully implicit"
method by reducing the complexity of the implicit algebraic equations that must
be solved during each integration step. Semi-implicit differencing is particularly
attractive when all the terms that are evaluated implicitly are linear functions of
the unknown variables.
In order to investigate the stability of semi-implicit time-differencing schemes,
consider a prototype ordinary differential equation of the form
(7.23)
This is simply aversion of the oscillation equation (2.30) in which the oscillatory
forcing is divided into high -frequency (WH) and low -frequency (WL) components.
The division of the forcing into two terms may appear to be rather artificial, but the