2.3 Time-Differencing
47
and the nonnegativity of 6.t and Sx implies
e6.t/6.x O.
(2.28)
Simultaneous satisfaction of (2.27) and (2.28) is obtained when
6.t
0< e - < 1.
- 6.xIn the case c < 0, similar reasoning leads to contradictory requirements, and the
solution is unstable.
The preceding stability condition is identical to those already obtained using the
energy and Von Neumann methods, but such agreement is actually rather unusual.
The CFL condition is only a necessary condition for stability, and in many cases
the sufficient conditions for stability are more restrictive than those required by
the CFL condition. As an example, consider the following approximation to the
advection equation,
which uses the fourth-order accurate approximation to the spatial derivative (2.6).
Since the spatial difference utilizes a five-grid-point-wide stencil, the CFL condition is satisfied when
Yet the actual sufficient condition for stability is the much more restrictive condition
l e Is 0.728,
which may be derived via a Von Neumann stability analysis.
2.3 Time-Differencing
A given partial differential equation can be approximated by an almost unlimited
variety of different finite-difference formulae. In order to systematically examine the properties of various finite-difference schemes, let us begin by discussing
possible approximations to the time derivative without explicitly considering the
spatial derivatives . The primary reason for discussing time and space differencing
separately is that they present the numerical analyst with rather different sets of
practical problems. After the nth step of the integration, the numerical solution
may be easily included in any finite-difference approximation to the spatial derivatives. It is easy, for example, to construct high-order centered approximations to
47
and the nonnegativity of 6.t and Sx implies
e6.t/6.x O.
(2.28)
Simultaneous satisfaction of (2.27) and (2.28) is obtained when
6.t
0< e - < 1.
- 6.xIn the case c < 0, similar reasoning leads to contradictory requirements, and the
solution is unstable.
The preceding stability condition is identical to those already obtained using the
energy and Von Neumann methods, but such agreement is actually rather unusual.
The CFL condition is only a necessary condition for stability, and in many cases
the sufficient conditions for stability are more restrictive than those required by
the CFL condition. As an example, consider the following approximation to the
advection equation,
which uses the fourth-order accurate approximation to the spatial derivative (2.6).
Since the spatial difference utilizes a five-grid-point-wide stencil, the CFL condition is satisfied when
Yet the actual sufficient condition for stability is the much more restrictive condition
l e Is 0.728,
which may be derived via a Von Neumann stability analysis.
2.3 Time-Differencing
A given partial differential equation can be approximated by an almost unlimited
variety of different finite-difference formulae. In order to systematically examine the properties of various finite-difference schemes, let us begin by discussing
possible approximations to the time derivative without explicitly considering the
spatial derivatives . The primary reason for discussing time and space differencing
separately is that they present the numerical analyst with rather different sets of
practical problems. After the nth step of the integration, the numerical solution
may be easily included in any finite-difference approximation to the spatial derivatives. It is easy, for example, to construct high-order centered approximations to