8.1 One-Dimensional Flow
409
8.1.6 Stability in the Presence 0/Boundaries
Pure initial-value, or Cauchy, problems are posed on infinite or periodic domains.
Methods for analyzing the stability of finite-difference approximations to Cauchy
problems were presented in Section 2.2. Unfortunately, the imposition of a boundary condition at the edge of a limited domain can destabilize numerical methods
that are stable approximations to the Cauchy problem . In this section we will examine the additional properties that must be satisfied to ensure the stability of
numerical approximations to initial-boundary -value problems (IBVP).
As in Section 2.2, the present analysis will be restricted to linear problems with
constant coefficients, for which the theory is simplest and most complete . The
easiest way to determine the stability of a linear constant-coefficient initial value
problem on the unbounded domain -00 < x < 00 is to use Von Neumann 's
method, which examines the behavior of individual waves of the form
ei(kjÖx-wnÖt) •
(8.20)
According to the Von Neumann criterion, a numerical method is stable if it does
not amplify any of the modes resolved on the numerical mesh. Defining K =
e i kÖx and z = e- i w 6./ , the grid-point values associated with each wave may be
written Kj z" . The Von Neumann stability of the Cauchy problem requires lzI I
for all IK 1 = 1. Spatial distributions other than pure sinusoidal waves mayaIso
be represented in the form K j z" . If IK 1 < I, the expression (8.20) represents
an oscillation whose amplitude grows exponentially as x
-00. There is no
need to worry about the stability of this type of mode in the Cauchy problem,
because it is not an admissible solution on a periodic or unbounded domain. If,
however, the same equations are to be solved as an IBVP on the half-infinite
domain x 2: 0, then the mode is admissible (since IKlj
0 as j
(0). The
stability requirement for IBVP on the half-bounded domain x 2: 0 must therefore
be extended to the Godunov-Ryabenki condition that Izi
I for all IKI
1.
(If the domain is -00 < x
0, the modes to be examined are those for which
IKI 2: 1.)
Although the Godunov-Ryabenki condition is necessary for stability, it is not
sufficient to guarantee that numerical solutions to IBVP are stable. General conditions for stability were given by Gustafsson , Kreiss, and Sundström (1972), who
noted that in practice , the most important stability question involves the behavior
of modes for which lz1= land IK 1= 1. These modes are undamped waves; they
can lead to instability if the interior finite-difference scheme together with the numerical boundary condition permits unforced waves to propagate inward through
the boundary. Trefethen (1983) has shown that the Gustafsson-Kreiss-Sundström
(GKS) stability condition is essentially equivalent to the requirement that (1) the
interior difference formula be stable for the Cauchy problem; (2) the model (including the boundary conditions) admit no eigensolutions that amplify with each
time step by a constant factor z with lzl > 1 (i.e., the Godunov-Ryabenki stability condition is satisfied), and (3) the model (including the boundary condi-
409
8.1.6 Stability in the Presence 0/Boundaries
Pure initial-value, or Cauchy, problems are posed on infinite or periodic domains.
Methods for analyzing the stability of finite-difference approximations to Cauchy
problems were presented in Section 2.2. Unfortunately, the imposition of a boundary condition at the edge of a limited domain can destabilize numerical methods
that are stable approximations to the Cauchy problem . In this section we will examine the additional properties that must be satisfied to ensure the stability of
numerical approximations to initial-boundary -value problems (IBVP).
As in Section 2.2, the present analysis will be restricted to linear problems with
constant coefficients, for which the theory is simplest and most complete . The
easiest way to determine the stability of a linear constant-coefficient initial value
problem on the unbounded domain -00 < x < 00 is to use Von Neumann 's
method, which examines the behavior of individual waves of the form
ei(kjÖx-wnÖt) •
(8.20)
According to the Von Neumann criterion, a numerical method is stable if it does
not amplify any of the modes resolved on the numerical mesh. Defining K =
e i kÖx and z = e- i w 6./ , the grid-point values associated with each wave may be
written Kj z" . The Von Neumann stability of the Cauchy problem requires lzI I
for all IK 1 = 1. Spatial distributions other than pure sinusoidal waves mayaIso
be represented in the form K j z" . If IK 1 < I, the expression (8.20) represents
an oscillation whose amplitude grows exponentially as x
-00. There is no
need to worry about the stability of this type of mode in the Cauchy problem,
because it is not an admissible solution on a periodic or unbounded domain. If,
however, the same equations are to be solved as an IBVP on the half-infinite
domain x 2: 0, then the mode is admissible (since IKlj
0 as j
(0). The
stability requirement for IBVP on the half-bounded domain x 2: 0 must therefore
be extended to the Godunov-Ryabenki condition that Izi
I for all IKI
1.
(If the domain is -00 < x
0, the modes to be examined are those for which
IKI 2: 1.)
Although the Godunov-Ryabenki condition is necessary for stability, it is not
sufficient to guarantee that numerical solutions to IBVP are stable. General conditions for stability were given by Gustafsson , Kreiss, and Sundström (1972), who
noted that in practice , the most important stability question involves the behavior
of modes for which lz1= land IK 1= 1. These modes are undamped waves; they
can lead to instability if the interior finite-difference scheme together with the numerical boundary condition permits unforced waves to propagate inward through
the boundary. Trefethen (1983) has shown that the Gustafsson-Kreiss-Sundström
(GKS) stability condition is essentially equivalent to the requirement that (1) the
interior difference formula be stable for the Cauchy problem; (2) the model (including the boundary conditions) admit no eigensolutions that amplify with each
time step by a constant factor z with lzl > 1 (i.e., the Godunov-Ryabenki stability condition is satisfied), and (3) the model (including the boundary condi-