396
8. Nonreflecting Boundary Conditions
(1949, p. 189), who defined it as the condition that "the sources must be sourees,
not sinks, of energy. The energy which is radiated from the sources must seatter
to infinity; no energy may be radiated from infinity into ... the field ." As formulated by Sommerfeld, the radiation eondition applies at infinity; however, in all
praetical eomputations a boundary condition must be imposed at some finite distance from the energy source , and this ereates two problems. The first problem is
that the radiation eondition itself may not properly describe the physical behavior
occurring at an arbitrarily designated location within the fluid when that loeation
is only a finite distance from the energy source . The seeond problem is that it
is typieally more difficult to express the radiation eondition mathematically at a
boundary that is only a finite distanee from the energy souree.
The radiation condition is obviously not appropriate in situations where energy
must be transmitted inward through the boundary; yet inward radiation may even
be required in problems where the surrounding fluid is initially quieseent and all
the initial disturbances are eontained within the eomputational domain. Two nonlinear waves that propagate past an artificial boundary within a fluid may interaet
outside the artificial boundary and generate an inward-propagating disturbanee
that should reenter the domain. This point was emphasized by Hedstrom (1979b),
who provided a simple example from eompressible gas dynamics demonstrating
that a shoek overtaking a contact diseontinuity must generate an echo that propagates back toward the wave generator. Numerieal simulations of thunderstorms
provide another example where the doeumented sensitivity of numerical simulations to the lateral boundary eonditions (Clark 1979; Hedley and Yau 1988) may
be not only a eonsequenee of poorly approximating the radiation eondition, but
also the result of inadequately representing important feedbacks on the conveetion arising through interaetions with the storm's environment that oecur outside
the numerieal domain .
Errors resulting from a failure to incorporate inward-propagating signals generated by real physical proeesses oecurring outside the boundaries of the computational domain eannot be avoided without enlarging the domain . Nevertheless,
nonreflecting boundary eonditions often beeome reasonable approximations to
the true physical boundary eondition as the size of the eomputational domain increases, provided that the loeal energy density of a disturbanee arriving at the
boundary is redueed as the result of wave dispersion or absorption in the large domain . When the disturbanees arriving at the boundary are sufficiently weak, the
goveming equations in the region near the boundary ean beapproximated by their
linearized equivalents, and it ean be relatively easy to ensure that the radiation
eondition eorreetly deseribes the boundary eonditions for the linearized system.
The situation is particularly simple in the case of eonstant-eoefficient linear hyperbolic systems, for which radiation boundary conditions are clearly appropriate
because the eharacteristic curves for sueh systems are straight lines that eannot
exit and subsequently reenter the domain .
Even when it is clear that the radiation eondition is appropriate, it is not always
easy to translate Sommerfeld's physical deseription into a mathematieal formula.
As noted in the monograph by Givoli (1992), it is generally easier to express the
8. Nonreflecting Boundary Conditions
(1949, p. 189), who defined it as the condition that "the sources must be sourees,
not sinks, of energy. The energy which is radiated from the sources must seatter
to infinity; no energy may be radiated from infinity into ... the field ." As formulated by Sommerfeld, the radiation eondition applies at infinity; however, in all
praetical eomputations a boundary condition must be imposed at some finite distance from the energy source , and this ereates two problems. The first problem is
that the radiation eondition itself may not properly describe the physical behavior
occurring at an arbitrarily designated location within the fluid when that loeation
is only a finite distance from the energy source . The seeond problem is that it
is typieally more difficult to express the radiation eondition mathematically at a
boundary that is only a finite distanee from the energy souree.
The radiation condition is obviously not appropriate in situations where energy
must be transmitted inward through the boundary; yet inward radiation may even
be required in problems where the surrounding fluid is initially quieseent and all
the initial disturbances are eontained within the eomputational domain. Two nonlinear waves that propagate past an artificial boundary within a fluid may interaet
outside the artificial boundary and generate an inward-propagating disturbanee
that should reenter the domain. This point was emphasized by Hedstrom (1979b),
who provided a simple example from eompressible gas dynamics demonstrating
that a shoek overtaking a contact diseontinuity must generate an echo that propagates back toward the wave generator. Numerieal simulations of thunderstorms
provide another example where the doeumented sensitivity of numerical simulations to the lateral boundary eonditions (Clark 1979; Hedley and Yau 1988) may
be not only a eonsequenee of poorly approximating the radiation eondition, but
also the result of inadequately representing important feedbacks on the conveetion arising through interaetions with the storm's environment that oecur outside
the numerieal domain .
Errors resulting from a failure to incorporate inward-propagating signals generated by real physical proeesses oecurring outside the boundaries of the computational domain eannot be avoided without enlarging the domain . Nevertheless,
nonreflecting boundary eonditions often beeome reasonable approximations to
the true physical boundary eondition as the size of the eomputational domain increases, provided that the loeal energy density of a disturbanee arriving at the
boundary is redueed as the result of wave dispersion or absorption in the large domain . When the disturbanees arriving at the boundary are sufficiently weak, the
goveming equations in the region near the boundary ean beapproximated by their
linearized equivalents, and it ean be relatively easy to ensure that the radiation
eondition eorreetly deseribes the boundary eonditions for the linearized system.
The situation is particularly simple in the case of eonstant-eoefficient linear hyperbolic systems, for which radiation boundary conditions are clearly appropriate
because the eharacteristic curves for sueh systems are straight lines that eannot
exit and subsequently reenter the domain .
Even when it is clear that the radiation eondition is appropriate, it is not always
easy to translate Sommerfeld's physical deseription into a mathematieal formula.
As noted in the monograph by Givoli (1992), it is generally easier to express the