2.2 Stabilityand Convergence
41
solution could hardly be expected to converge to the true solution in the limit I::!.x ,
The practical shortcoming of the preceding definition of stability is that it says
nothing about the quality of the solution that might be obtained when using finite
values of I::!.t and I::!.x ; it only ensures that an accurate solution will be obtained
in the limit S», I::!.t
O. Schemes that are stable according to the criteria (2.15)
may, nevertheless, generate solutions that "blow up" in practical applications (see
Section 3.4.3 for an example). In order to ensure that the numerical solution is
qualitatively similar to the true solution when I::!.x and I::!.t are finite, it is often
useful to impose stability constraints that are more stringent than (2.15). In many
wave propagation problems, the norm of the true solution is constant with time,
and in such instances it is appropriate to require that the numerical scheme satisfy
(2.16)
In contrast to (2.15), this condition is not necessary for convergence, and it cannot
be sensibly imposed without specific knowledge about the boundedness of the
solutions to the associated partial differential equation. Nevertheless, (2.16) is a
perfectly reasonable constraint to impose in applications where the true solution
is not growing with time, and unlike (2.15), it guarantees that the solution will not
blow up.
It is relatively easy to formulate consistent difference schemes and to determine
their truncation error and order of accuracy. The analysis of stability can, however,
be far more difficult, particularly when the finite-difference scheme and the associated partial differential equation are nonlinear. Thus, our initial discussion of
stability will be focused on the simplest case-Iinear finite-difference schemes
for the approximation of linear partial differential equations with constant coefficients. Nonlinear equations and linear equations with variable coefficients will be
considered in Chapter 3.
2.2.1 The Energy Method
In practice, the energy method is used much less frequently than the Von Neumann
method, which will be discussed in the next section. Nevertheless, the energy
method is important, because unlike the Von Neumann method, it can be applied
to nonlinear equations and to problems without periodic boundaries. The basic
idea behind the energy method is to find a posit ive definite quantity like Lj (4J'j)2
and show that this quantity is bounded for all n. If Lj(4J'j)2 is bounded, the
solution is stable with respect to the l2-norm.
As an example, let us investigate the stability of the upstream finite-difference
scheme (2.11) . Defining J.t = cI::!.t / I::!.x, the scheme may be written as
(2.17)
41
solution could hardly be expected to converge to the true solution in the limit I::!.x ,
The practical shortcoming of the preceding definition of stability is that it says
nothing about the quality of the solution that might be obtained when using finite
values of I::!.t and I::!.x ; it only ensures that an accurate solution will be obtained
in the limit S», I::!.t
O. Schemes that are stable according to the criteria (2.15)
may, nevertheless, generate solutions that "blow up" in practical applications (see
Section 3.4.3 for an example). In order to ensure that the numerical solution is
qualitatively similar to the true solution when I::!.x and I::!.t are finite, it is often
useful to impose stability constraints that are more stringent than (2.15). In many
wave propagation problems, the norm of the true solution is constant with time,
and in such instances it is appropriate to require that the numerical scheme satisfy
(2.16)
In contrast to (2.15), this condition is not necessary for convergence, and it cannot
be sensibly imposed without specific knowledge about the boundedness of the
solutions to the associated partial differential equation. Nevertheless, (2.16) is a
perfectly reasonable constraint to impose in applications where the true solution
is not growing with time, and unlike (2.15), it guarantees that the solution will not
blow up.
It is relatively easy to formulate consistent difference schemes and to determine
their truncation error and order of accuracy. The analysis of stability can, however,
be far more difficult, particularly when the finite-difference scheme and the associated partial differential equation are nonlinear. Thus, our initial discussion of
stability will be focused on the simplest case-Iinear finite-difference schemes
for the approximation of linear partial differential equations with constant coefficients. Nonlinear equations and linear equations with variable coefficients will be
considered in Chapter 3.
2.2.1 The Energy Method
In practice, the energy method is used much less frequently than the Von Neumann
method, which will be discussed in the next section. Nevertheless, the energy
method is important, because unlike the Von Neumann method, it can be applied
to nonlinear equations and to problems without periodic boundaries. The basic
idea behind the energy method is to find a posit ive definite quantity like Lj (4J'j)2
and show that this quantity is bounded for all n. If Lj(4J'j)2 is bounded, the
solution is stable with respect to the l2-norm.
As an example, let us investigate the stability of the upstream finite-difference
scheme (2.11) . Defining J.t = cI::!.t / I::!.x, the scheme may be written as
(2.17)