32
1. Introduction
Physicists seldom worry about we11-posedness, since properly formulated
mathematical models of the physical world are almost always well-posed. The
preceding example may be recognized as an initial value problem in which y
represents time and x is the spatial coordinate. In contrast to their hyperbolic
cousins, elliptic partial differential equations describe steady-state physical systems and do not natura11y arise as initial value problems. When a real-world system is govemed by an elliptic equation, physical considerations usua11y provide
data for the dependent variables or their normal derivatives along each boundary, and the additional boundary-value data leads to a we11-posed problem. The
fact that elliptic partial differential equations are not we11-posed as initial value
problems may therefore be irrelevant to the physicist-but it is not irrelevant to
the numerical analyst. Given a well-posed elliptic problem, such as (1.69) with
y < 0 and u specified at y = 0 and y = Y, could one expect to compute an
accurate approximate solution on some numerical grid by starting with the known
values along one boundary and stepping across the grid, one point at a time? The
answer is no, an approach of this type is numerica11y unstable-indeed it mimics the not-we11-posed formulation of an elliptic partial differential equation as
an initial value problem. Practical methods for the numerical solution of elliptic partial differential equations are therefore not " marching" schemes . Instead
of computing the solution at one point and then proceeding to the next, a11 the
grid-point values must be simultaneously adjusted (perhaps through some iterative process) in order to adequately satisfy the goveming differential equation and
the boundary conditions. In contrast, hyperbolic partial differential equations do
lend themselves to numerical solution via marehing techniques.l
Another major difference in the numerical treatment of elliptic and hyperbolic
equations arises in the specification of boundary conditions. As suggested by the
preceding example, boundary conditions are usua11y imposed at every boundary
as part of the natural formulation of an elliptic problem. Moreover, the incorporation of these boundary data into a numerical algorithm is generally straightforward. On the other hand, if one attempts to compute the solution to a hyperbolic
problem in a limited spatial domain, the numerical algorithm may require boundary conditions in regions where none should actua11y be specified (i.e., at a boundary where a11 the characteristic curves are directed out of the domain). Improper
boundary conditions may lead to instabilities or to nonuniqueness in the numerical solution of a hyperbolic system . Further discussion of boundary conditions
will be presented in Chapter 8.
5L.F. Richardson, who explored the numerical solution of a variety of partial differential equations prior to his celebrated attempt at numerical weather prediction, coined the terms "jury" and
"marching " methods to describe the basic difference between the numerical techniques suitable for
the solution of elliptic equations and hyperbolic equations . The adjective "jury" alluded to the idea
that one needed to adjust all the values in the numerical solution until the whole was "judged" to
constitute a satisfactory approximation .
1. Introduction
Physicists seldom worry about we11-posedness, since properly formulated
mathematical models of the physical world are almost always well-posed. The
preceding example may be recognized as an initial value problem in which y
represents time and x is the spatial coordinate. In contrast to their hyperbolic
cousins, elliptic partial differential equations describe steady-state physical systems and do not natura11y arise as initial value problems. When a real-world system is govemed by an elliptic equation, physical considerations usua11y provide
data for the dependent variables or their normal derivatives along each boundary, and the additional boundary-value data leads to a we11-posed problem. The
fact that elliptic partial differential equations are not we11-posed as initial value
problems may therefore be irrelevant to the physicist-but it is not irrelevant to
the numerical analyst. Given a well-posed elliptic problem, such as (1.69) with
y < 0 and u specified at y = 0 and y = Y, could one expect to compute an
accurate approximate solution on some numerical grid by starting with the known
values along one boundary and stepping across the grid, one point at a time? The
answer is no, an approach of this type is numerica11y unstable-indeed it mimics the not-we11-posed formulation of an elliptic partial differential equation as
an initial value problem. Practical methods for the numerical solution of elliptic partial differential equations are therefore not " marching" schemes . Instead
of computing the solution at one point and then proceeding to the next, a11 the
grid-point values must be simultaneously adjusted (perhaps through some iterative process) in order to adequately satisfy the goveming differential equation and
the boundary conditions. In contrast, hyperbolic partial differential equations do
lend themselves to numerical solution via marehing techniques.l
Another major difference in the numerical treatment of elliptic and hyperbolic
equations arises in the specification of boundary conditions. As suggested by the
preceding example, boundary conditions are usua11y imposed at every boundary
as part of the natural formulation of an elliptic problem. Moreover, the incorporation of these boundary data into a numerical algorithm is generally straightforward. On the other hand, if one attempts to compute the solution to a hyperbolic
problem in a limited spatial domain, the numerical algorithm may require boundary conditions in regions where none should actua11y be specified (i.e., at a boundary where a11 the characteristic curves are directed out of the domain). Improper
boundary conditions may lead to instabilities or to nonuniqueness in the numerical solution of a hyperbolic system . Further discussion of boundary conditions
will be presented in Chapter 8.
5L.F. Richardson, who explored the numerical solution of a variety of partial differential equations prior to his celebrated attempt at numerical weather prediction, coined the terms "jury" and
"marching " methods to describe the basic difference between the numerical techniques suitable for
the solution of elliptic equations and hyperbolic equations . The adjective "jury" alluded to the idea
that one needed to adjust all the values in the numerical solution until the whole was "judged" to
constitute a satisfactory approximation .