412
8. NonreflectingBoundary Conditions
The only resolvable modes satisfying the boundary condition (8.23) are those for
which kfix = O. The group velocities of all such modes have the same sign as U
and are directed outward through the downstream boundary. Since the numerical
scheme does not support free modes with inward-directed group velocities, it is
stable.
Observe that the extrapolation condition 4>;; = 4>1 would never be stable for
an inflow boundary, because the group velocity of the zero-wave-number physical mode (eo, k) = (0,0) is directed inward through the boundary, and this mode
satisfies both the boundary condition and the interior finite-difference scheme.
Such instability might have been anticipated from the reflection-coefficient analysis in the preceding section, which indicated that the reflection coefficient associated with zero-order extrapolation at an inflow boundary becomes infinite as
k S» -+ tt . The connection between the reflection coefficient and stability is,
however, somewhat complex. Boundary conditions associated with infinite coefficients are always unstable, but GKS stability does not require IrI 1. Further
discussion of the relation between reflection coefficients and instability appears in
Trefethen (1985). The stability, or lack thereof, of several basic methods for the
numerical solution of the advection equation is presented in aseries of examples
by Goldberg and Tadmor (1985).
In most situations, two boundaries are present, and one needs stable methods
for closed spatial domains such as 0
x
L . Gustafsson et al. (1972) showed
that the stability of the two-boundary problem can be determined by analyzing
each boundary separately. If the boundary condition at x = 0, and the interior
finite-difference formulae, are GKS-stable approximations to an IBVP on the domain 0
x < 00, and if the boundary condition at x = L is also GKS stable
for problems on the domain -00 < x
0, then the two-boundary problem is
GKS stable. The basic reason that each boundary can be considered separately
is that the simultaneous presence of two boundaries does not admit new types
of eigensolutions beyond those whose stability was already tested in the pair of
single-boundary problems. Nevertheless, GKS stability does not guarantee that
the solution to the two-boundary problem will be completely satisfactory. In particular, it is possible for a method to be GKS stable but to generate reflections
with IrI > 1. If the reflection coefficient at both boundaries exceeds unity, the
solution may grow with time. Such growth is not a true instability, in the sense
that it need not prevent the convergence of integrations performed over some fixed
time interval, since in theory, the error can be made arbitrarily small in the limit
fix -+ 0, fit -+ O. Nevertheless, in most practical situations it is advisable to
choose a method with IrI I at both boundaries .
8.2 Two-Dimensional Shallow-Water Flow
Exact and practically useful mathematical formulae for the specification of nonreflecting boundary conditions are seldom available in multidimensional prob-
8. NonreflectingBoundary Conditions
The only resolvable modes satisfying the boundary condition (8.23) are those for
which kfix = O. The group velocities of all such modes have the same sign as U
and are directed outward through the downstream boundary. Since the numerical
scheme does not support free modes with inward-directed group velocities, it is
stable.
Observe that the extrapolation condition 4>;; = 4>1 would never be stable for
an inflow boundary, because the group velocity of the zero-wave-number physical mode (eo, k) = (0,0) is directed inward through the boundary, and this mode
satisfies both the boundary condition and the interior finite-difference scheme.
Such instability might have been anticipated from the reflection-coefficient analysis in the preceding section, which indicated that the reflection coefficient associated with zero-order extrapolation at an inflow boundary becomes infinite as
k S» -+ tt . The connection between the reflection coefficient and stability is,
however, somewhat complex. Boundary conditions associated with infinite coefficients are always unstable, but GKS stability does not require IrI 1. Further
discussion of the relation between reflection coefficients and instability appears in
Trefethen (1985). The stability, or lack thereof, of several basic methods for the
numerical solution of the advection equation is presented in aseries of examples
by Goldberg and Tadmor (1985).
In most situations, two boundaries are present, and one needs stable methods
for closed spatial domains such as 0
x
L . Gustafsson et al. (1972) showed
that the stability of the two-boundary problem can be determined by analyzing
each boundary separately. If the boundary condition at x = 0, and the interior
finite-difference formulae, are GKS-stable approximations to an IBVP on the domain 0
x < 00, and if the boundary condition at x = L is also GKS stable
for problems on the domain -00 < x
0, then the two-boundary problem is
GKS stable. The basic reason that each boundary can be considered separately
is that the simultaneous presence of two boundaries does not admit new types
of eigensolutions beyond those whose stability was already tested in the pair of
single-boundary problems. Nevertheless, GKS stability does not guarantee that
the solution to the two-boundary problem will be completely satisfactory. In particular, it is possible for a method to be GKS stable but to generate reflections
with IrI > 1. If the reflection coefficient at both boundaries exceeds unity, the
solution may grow with time. Such growth is not a true instability, in the sense
that it need not prevent the convergence of integrations performed over some fixed
time interval, since in theory, the error can be made arbitrarily small in the limit
fix -+ 0, fit -+ O. Nevertheless, in most practical situations it is advisable to
choose a method with IrI I at both boundaries .
8.2 Two-Dimensional Shallow-Water Flow
Exact and practically useful mathematical formulae for the specification of nonreflecting boundary conditions are seldom available in multidimensional prob-