8.1 One-Dimensional Flow
397
radiation condition at infinity mathematically than to formulate the same condition for a boundary that is only a finite distance from the wave source . In fact, in
most practical problems it is impossible to express the radiation condition exactly
as an algebraic or differential equation involving the prognostic variables in the
neighborhood of the boundary. Instead of the exact radiation condition one must
use an approximation, and this approximation introduces an error in the mathematical model of the physical system that is distinct from the errors subsequently
incurred when constructing a discrete approximation to the mathematical model.
As a consequence, the numerical solution will not converge to the correct solution to the underlying physical problem as the spatial and temporal mesh is refined, but rather to the correct solution to the approximate mathematical problem
deterrnined by the approximate radiation condition.
The first topic that will be considered in this chapter is therefore the mathematical forrnulation of well-posed radiation boundary conditions . We begin with
examples where this can be done exactly and then consider problems where approximations are required. After forrnulating exact or approximate radiation
boundary conditions for the continuous problem , we will consider their numerical
implementation.
8.1 One-Dimensional Flow
Exact open boundary conditions can be obtained for certain simple one-dimensional systems. Two such problems will be considered in this section, the linear
advection equation and the linearized shallow-water system. The shallow-water
system provides an instructive example illustrating the construction of exact radiation boundary conditions for the continuous equations. In contrast, there is no
need to explicitly deterrnine a radiation boundary condition for the nondiscretized
linear advection equation because a well-posed mathematical forrnulation of the
advection problem does not require any outftow boundary condition. The linear
advection equation is, however, useful for investigating the inftuence of various
numerical approximations to the exact outftow boundary condition on the accuracy and stability of the discretized solution .
8.1.1 Well-Posed Initial-Boundary Value Problems.
Consider the linearized one-dimensional advection equation
a1/l
a1/l
- + U - = O,
at
ax
(8.1)
and suppose 1/1 is to be determined throughout some limited domain 0 ::: x ::: L.
For the sake of illustration, assurne that U > O. If initial data are given for the
397
radiation condition at infinity mathematically than to formulate the same condition for a boundary that is only a finite distance from the wave source . In fact, in
most practical problems it is impossible to express the radiation condition exactly
as an algebraic or differential equation involving the prognostic variables in the
neighborhood of the boundary. Instead of the exact radiation condition one must
use an approximation, and this approximation introduces an error in the mathematical model of the physical system that is distinct from the errors subsequently
incurred when constructing a discrete approximation to the mathematical model.
As a consequence, the numerical solution will not converge to the correct solution to the underlying physical problem as the spatial and temporal mesh is refined, but rather to the correct solution to the approximate mathematical problem
deterrnined by the approximate radiation condition.
The first topic that will be considered in this chapter is therefore the mathematical forrnulation of well-posed radiation boundary conditions . We begin with
examples where this can be done exactly and then consider problems where approximations are required. After forrnulating exact or approximate radiation
boundary conditions for the continuous problem , we will consider their numerical
implementation.
8.1 One-Dimensional Flow
Exact open boundary conditions can be obtained for certain simple one-dimensional systems. Two such problems will be considered in this section, the linear
advection equation and the linearized shallow-water system. The shallow-water
system provides an instructive example illustrating the construction of exact radiation boundary conditions for the continuous equations. In contrast, there is no
need to explicitly deterrnine a radiation boundary condition for the nondiscretized
linear advection equation because a well-posed mathematical forrnulation of the
advection problem does not require any outftow boundary condition. The linear
advection equation is, however, useful for investigating the inftuence of various
numerical approximations to the exact outftow boundary condition on the accuracy and stability of the discretized solution .
8.1.1 Well-Posed Initial-Boundary Value Problems.
Consider the linearized one-dimensional advection equation
a1/l
a1/l
- + U - = O,
at
ax
(8.1)
and suppose 1/1 is to be determined throughout some limited domain 0 ::: x ::: L.
For the sake of illustration, assurne that U > O. If initial data are given for the