408
8. Nonreflecting Boundary Conditions
The time derivative was approximated using trapezoidal time-differencing with
a small Courant number. The initial condition is shown in Fig. 8.2a. Figure 8.2b
shows the solution shortly before the disturbances encounter the lateral boundaries; the direction in which each disturbance is propagating is indicated by an
arrow. The long-wave component of the solution travels toward the outflow boundary in a physically correct manner, but the 2ßx wave packet propagates upwind
at speed -U. After the disturbances have reflected off the lateral boundaries, the
solution appears as shown in Fig. 8.2c. Physical modes are reflected into compu -
tational modes by the outflow boundary condition, but since the initial disturbance
is well-resolved, the reflection is relatively weak. The left-moving computational
mode is reflected, without loss of amplitude, into a right -moving physical mode at
the inflow boundary. The strong reflection at the right boundary is not necessarily
indicative of a deficiency in the inflow boundary condition. Only computational
modes lead to difficulties at the inflow boundary, and even those modes are not
amplified . One can eliminate the problem at the upstream boundary by filtering
the computational modes out of the interior solution.
In addition to the evaluation of the reflection coefficient, one can also examine
the formal accuracy of the numerical boundary condition by substituting Taylor series expansions into the discretized formula in the usual manner (see Section 2.1). Gustafsson (1975) has shown that there is generally no degradation in
the overall order of convergence if the boundary conditions are formulated with
one order less accuracy than that used for the interior finite differences. In many
practical problems the order of accuracy of the discretized approximation to the
radiation condition is actually a relatively minor concern . The numerical errors
generated at the boundary often involve reflections into poorly resolved short
waves, and the accuracy achieved in the representation of such poorly resolved
features is not reliably determined by the leading -order term in the truncation error. Moreover, in complex multidimensional problems the most serious errors are
often introduced in the mathematical formulation of approximate radiation boundary conditions, and it is not necessary to implement these approximate conditions
with highly accurate numerical formulae.
In fact, the optimal nonreflecting boundary condition for a discretized problem
is not generally a high-order approximation to the exact one-way wave equation
for the continuous problem, but rather the one-way wave equation whose discrete
dispersion relation best replicates the propagation characteristics of the interior
finite-difference scheme (Engquist and Majda 1979). As an example, suppose that
solutions are sought to the one-dimensional advection equation using the semidiscrete approximation (8.14). The optimal boundary condition would be one that
masks the apparent change in the propagation medium generated by the difference between the finite-difference formulae at the boundary and the interior of
the domain . The dispersion relation for the one-way wave equation that will pass
solutions to this semidiscrete system through the boundary with zero reflection
would be w = U sin(kßx)j ßx rather than the exact condition w = Uk . Unfortunately, techniques are not currently available to create a practical boundary
condition with either of these dispersion relations.
8. Nonreflecting Boundary Conditions
The time derivative was approximated using trapezoidal time-differencing with
a small Courant number. The initial condition is shown in Fig. 8.2a. Figure 8.2b
shows the solution shortly before the disturbances encounter the lateral boundaries; the direction in which each disturbance is propagating is indicated by an
arrow. The long-wave component of the solution travels toward the outflow boundary in a physically correct manner, but the 2ßx wave packet propagates upwind
at speed -U. After the disturbances have reflected off the lateral boundaries, the
solution appears as shown in Fig. 8.2c. Physical modes are reflected into compu -
tational modes by the outflow boundary condition, but since the initial disturbance
is well-resolved, the reflection is relatively weak. The left-moving computational
mode is reflected, without loss of amplitude, into a right -moving physical mode at
the inflow boundary. The strong reflection at the right boundary is not necessarily
indicative of a deficiency in the inflow boundary condition. Only computational
modes lead to difficulties at the inflow boundary, and even those modes are not
amplified . One can eliminate the problem at the upstream boundary by filtering
the computational modes out of the interior solution.
In addition to the evaluation of the reflection coefficient, one can also examine
the formal accuracy of the numerical boundary condition by substituting Taylor series expansions into the discretized formula in the usual manner (see Section 2.1). Gustafsson (1975) has shown that there is generally no degradation in
the overall order of convergence if the boundary conditions are formulated with
one order less accuracy than that used for the interior finite differences. In many
practical problems the order of accuracy of the discretized approximation to the
radiation condition is actually a relatively minor concern . The numerical errors
generated at the boundary often involve reflections into poorly resolved short
waves, and the accuracy achieved in the representation of such poorly resolved
features is not reliably determined by the leading -order term in the truncation error. Moreover, in complex multidimensional problems the most serious errors are
often introduced in the mathematical formulation of approximate radiation boundary conditions, and it is not necessary to implement these approximate conditions
with highly accurate numerical formulae.
In fact, the optimal nonreflecting boundary condition for a discretized problem
is not generally a high-order approximation to the exact one-way wave equation
for the continuous problem, but rather the one-way wave equation whose discrete
dispersion relation best replicates the propagation characteristics of the interior
finite-difference scheme (Engquist and Majda 1979). As an example, suppose that
solutions are sought to the one-dimensional advection equation using the semidiscrete approximation (8.14). The optimal boundary condition would be one that
masks the apparent change in the propagation medium generated by the difference between the finite-difference formulae at the boundary and the interior of
the domain . The dispersion relation for the one-way wave equation that will pass
solutions to this semidiscrete system through the boundary with zero reflection
would be w = U sin(kßx)j ßx rather than the exact condition w = Uk . Unfortunately, techniques are not currently available to create a practical boundary
condition with either of these dispersion relations.