404
8. Nonreflecting Boundary Conditions
ous problem. The second difficulty is that most finite-difference forrnulae support
nonphysical computational modes that propagate in a direction opposite to the
correct physical solution and may thereby blur the distinction between inflow and
outflow boundaries.
In order to examine the effects of spatial discretization, suppose that the advection equation (8.1) is approximated by the differential-difference equation
d
dt
2ßx
at j = 0, .. . , N discrete grid points in the finite domain 0
x
L. The preceding differential-difference equation cannot be used to calculate rPo and rPN
because the centered difference cannot be evaluated at a boundary. Suppose that
V < 0, so x = L is an inflow boundary. Then a boundary condition 1/I(L, r) =
I(t) must be imposed to render the continuous problem complete and well-posed.
This same inflow boundary condition may be used to specify rPN, but the numerical calculation of the solution at the other boundary will be a problem. There is
no boundary condition in the forrnulation of the nondiscretized initial-boundaryvalue problem that can be used to specify rPo. Since 1/1(0, t) is determined by the
interior solution, one might try to estimate 1/1(0, t) by extrapolating the interior
solution outward to the boundary. The simplest extrapolation is
(8.15)
This, of course, is a zero-gradient condition, and based on the earlier discussion
of zero -gradient conditions in the continuous shallow-water problem, one might
expect it to produce reflection .
The numerical boundary condition (8.15) does indeed produce some reflection,
but the situation is fundamentally different from that in the shallow-water system. When c > IVI, the shallow-water equations support physical modes that
propagate both to the right and to the left. Specifying aulox = 0 or aT/lox = 0
generates reflection from the outgoing physical mode into the incoming physical mode because these conditions are inadequate approximations to the correct
open boundary conditions in the shallow-water system. In contrast to the shallowwater system, all physical solutions to the scalar advection equation (8.1) travel
in the same direction. Any reflections that occur when (8.15) is used as a boundary condition for the differential-difference equation (8.14) involve nonphysical
numerical modes.
In the remainder of this section we will examine the interactions that may occur between physical and nonphysical modes as a result of the numerical approximations that are made at an open boundary. This investigation will focus
on the simplest dynamical system in which such interactions occur-the problem
of scalar advection. A similar, though more tedious, analysis may be performed
for shallow-water flow after allowing for the fact that centered second-order finitedifference approximations to the spatial derivatives in the one-dimensional
shallow-water system support two physical and two nonphysical modes. Under
8. Nonreflecting Boundary Conditions
ous problem. The second difficulty is that most finite-difference forrnulae support
nonphysical computational modes that propagate in a direction opposite to the
correct physical solution and may thereby blur the distinction between inflow and
outflow boundaries.
In order to examine the effects of spatial discretization, suppose that the advection equation (8.1) is approximated by the differential-difference equation
d
dt
2ßx
at j = 0, .. . , N discrete grid points in the finite domain 0
x
L. The preceding differential-difference equation cannot be used to calculate rPo and rPN
because the centered difference cannot be evaluated at a boundary. Suppose that
V < 0, so x = L is an inflow boundary. Then a boundary condition 1/I(L, r) =
I(t) must be imposed to render the continuous problem complete and well-posed.
This same inflow boundary condition may be used to specify rPN, but the numerical calculation of the solution at the other boundary will be a problem. There is
no boundary condition in the forrnulation of the nondiscretized initial-boundaryvalue problem that can be used to specify rPo. Since 1/1(0, t) is determined by the
interior solution, one might try to estimate 1/1(0, t) by extrapolating the interior
solution outward to the boundary. The simplest extrapolation is
(8.15)
This, of course, is a zero-gradient condition, and based on the earlier discussion
of zero -gradient conditions in the continuous shallow-water problem, one might
expect it to produce reflection .
The numerical boundary condition (8.15) does indeed produce some reflection,
but the situation is fundamentally different from that in the shallow-water system. When c > IVI, the shallow-water equations support physical modes that
propagate both to the right and to the left. Specifying aulox = 0 or aT/lox = 0
generates reflection from the outgoing physical mode into the incoming physical mode because these conditions are inadequate approximations to the correct
open boundary conditions in the shallow-water system. In contrast to the shallowwater system, all physical solutions to the scalar advection equation (8.1) travel
in the same direction. Any reflections that occur when (8.15) is used as a boundary condition for the differential-difference equation (8.14) involve nonphysical
numerical modes.
In the remainder of this section we will examine the interactions that may occur between physical and nonphysical modes as a result of the numerical approximations that are made at an open boundary. This investigation will focus
on the simplest dynamical system in which such interactions occur-the problem
of scalar advection. A similar, though more tedious, analysis may be performed
for shallow-water flow after allowing for the fact that centered second-order finitedifference approximations to the spatial derivatives in the one-dimensional
shallow-water system support two physical and two nonphysical modes. Under