3.6 Nonlinear Instability
159
3.6 Nonlinear Instability
As discussed in the preceding section, the stability of finite-difference approximations to linear equations with variable coefficients can be detennined by examining the stability ofthe associated family of frozen-coefficient problems-provided
that the solution, and the variable coefficients, are sufficiently smooth and wellresolved on the numerical mesh. One may attempt to analyze the stability of nonlinear equations through a similar procedure. First, the non linear equations are
linearized; then a frozen-coefficient analysis is perfonned to detennine stability
conditions for the linearized problem. This approach gives necessary conditions
for stability, but as was the case with variable-coefficient linear equations, it may
give misleading results in situations where the solution is dominated by poorly
resolved short-wavelength perturbations. Unfortunately, the caveat that the solution must remain smooth and well-resolved is a much more serious impediment
to the analysis of nonlinear finite-difference equations because such equations can
rapidly generate unresolvable short-wave perturbations from very smooth initial
data.
In the following we will examine techniques for stabilizing the finite-difference
approximation of two nonlinear equations: Burgers 's equation and the barotropic
vorticity equation. Solutions to Burgers's equation often develop shocks and discontinuities whose accurate approximation requires the use of methods that will
be presented in Chapter 5. The schemes that will be considered in this section
provide very simple examples illustrating the stabilization of numerical approximations to a nonlinear problem by a judicious choice of finite-difference fonnula.
These schemes are not, however, recommended for practical applications involving the simulation of problems with shocks or discontinuous solutions. The opportunity for practical application of the ideas illustrated using Burgers 's equation
arises in attempting to avoid nonlinear instabilities in numerical solutions to the
barotropic vorticity equation. Solutions to the barotropic vorticity equation never
develop shocks and remain essentially as smooth as the initial data.
3.6.1 Burgers's Equation
The inviscid Burgers's equation,
a1/l + 1/1 a1/l = 0,
at
ax
(3.113)
is an example of a nonlinear partial differential equation whose solution rapidly
develops unresolvable short-wavelength components. If 1/I(x, 0) = !(x) at some
initial time t = 0, solutions to this problem can be written in the implicit form
1/1(x , t) = ! (x - 1/1(x , t)t) ,
which implies that ! is constant along the characteristic curves
x - 1/1(x , t)t = xo.
159
3.6 Nonlinear Instability
As discussed in the preceding section, the stability of finite-difference approximations to linear equations with variable coefficients can be detennined by examining the stability ofthe associated family of frozen-coefficient problems-provided
that the solution, and the variable coefficients, are sufficiently smooth and wellresolved on the numerical mesh. One may attempt to analyze the stability of nonlinear equations through a similar procedure. First, the non linear equations are
linearized; then a frozen-coefficient analysis is perfonned to detennine stability
conditions for the linearized problem. This approach gives necessary conditions
for stability, but as was the case with variable-coefficient linear equations, it may
give misleading results in situations where the solution is dominated by poorly
resolved short-wavelength perturbations. Unfortunately, the caveat that the solution must remain smooth and well-resolved is a much more serious impediment
to the analysis of nonlinear finite-difference equations because such equations can
rapidly generate unresolvable short-wave perturbations from very smooth initial
data.
In the following we will examine techniques for stabilizing the finite-difference
approximation of two nonlinear equations: Burgers 's equation and the barotropic
vorticity equation. Solutions to Burgers's equation often develop shocks and discontinuities whose accurate approximation requires the use of methods that will
be presented in Chapter 5. The schemes that will be considered in this section
provide very simple examples illustrating the stabilization of numerical approximations to a nonlinear problem by a judicious choice of finite-difference fonnula.
These schemes are not, however, recommended for practical applications involving the simulation of problems with shocks or discontinuous solutions. The opportunity for practical application of the ideas illustrated using Burgers 's equation
arises in attempting to avoid nonlinear instabilities in numerical solutions to the
barotropic vorticity equation. Solutions to the barotropic vorticity equation never
develop shocks and remain essentially as smooth as the initial data.
3.6.1 Burgers's Equation
The inviscid Burgers's equation,
a1/l + 1/1 a1/l = 0,
at
ax
(3.113)
is an example of a nonlinear partial differential equation whose solution rapidly
develops unresolvable short-wavelength components. If 1/I(x, 0) = !(x) at some
initial time t = 0, solutions to this problem can be written in the implicit form
1/1(x , t) = ! (x - 1/1(x , t)t) ,
which implies that ! is constant along the characteristic curves
x - 1/1(x , t)t = xo.