5
Finite-Volume Methods
As demonstrated in the preceding chapters, the errors in most numerical solutions increase dramatically as the physical scale of the simulated disturbance approaches the minimum scale resolvable on the numerical mesh . When solving
equations for which smooth initial data guarantees a smooth solution at all later
times, such as the barotropic vorticity equation (3.123), any difficulties associated
with poor numerical resolution can be avoided by using a sufficiently fine computational mesh. But if the goveming equations allow an initially smooth field to
develop shocks or discontinuities, as is the case with Burgers's equation (3.113),
there is no hope of maintaining adequate numerical resolution throughout the simulation, and special numerical techniques must be used to control the development
of overshoots and undershoots in the vicinity of the shock . Numerical approximations to equations with discontinuous solutions must also satisfy additional conditions beyond the stability and consistency requirements discussed in Chapter 2
to guarantee that the numerical solution converges to the correct solution as the
spatial grid interval and the time step approach zero.
The possibility of erroneous convergence to a function that does not approximate the true discontinuous solution can bedemonstrated by comparing numerical
solutions to the generalized Burgers 's equation in advective form
(5.1)
(5.2)
o1{l + 1{12 o1{l = 0
ot
OX
o1{l + (1{13) = o.
ot OX 3
with those generated by analogous solutions to the same equation influxform
D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999
Finite-Volume Methods
As demonstrated in the preceding chapters, the errors in most numerical solutions increase dramatically as the physical scale of the simulated disturbance approaches the minimum scale resolvable on the numerical mesh . When solving
equations for which smooth initial data guarantees a smooth solution at all later
times, such as the barotropic vorticity equation (3.123), any difficulties associated
with poor numerical resolution can be avoided by using a sufficiently fine computational mesh. But if the goveming equations allow an initially smooth field to
develop shocks or discontinuities, as is the case with Burgers's equation (3.113),
there is no hope of maintaining adequate numerical resolution throughout the simulation, and special numerical techniques must be used to control the development
of overshoots and undershoots in the vicinity of the shock . Numerical approximations to equations with discontinuous solutions must also satisfy additional conditions beyond the stability and consistency requirements discussed in Chapter 2
to guarantee that the numerical solution converges to the correct solution as the
spatial grid interval and the time step approach zero.
The possibility of erroneous convergence to a function that does not approximate the true discontinuous solution can bedemonstrated by comparing numerical
solutions to the generalized Burgers 's equation in advective form
(5.1)
(5.2)
o1{l + 1{12 o1{l = 0
ot
OX
o1{l + (1{13) = o.
ot OX 3
with those generated by analogous solutions to the same equation influxform
D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999