48
2. BasicFinite-Difference Methods
spatial derivatives. In contrast, storage limitations dictate that tP be retained at as
few time levels as possible, and the only time levels available are those from previous iterations . Thus , higher-order finite-difference approximations to the time
derivative are inherently one-sided.
The following discussion of the effects of time-differencing on the numerical
solution is transferable, after minor modification, to situations where the spatial
derivatives are approximated by centered differences. In addition, this discussion
provides an exact analysis of the influence of time-differencing on schemes, such
as the spectral method, where finite differences are not used to evaluate the spatial derivatives. Nevertheless, one must be careful not to assume that space and
time differences are completely independent. Indeed, techn iques such as the LaxWendroff method cannot be properly analyzed without understanding the interaction between space truncation error and time truncation error. The combined
effects of space- and time-differencing will be discussed, together with schemes
like the Lax-Wendroff method, in Section 2.5.
Time-differencing formulae used in the numerical solution of partial differential equations are related, naturally enough, to the numerical methods used to
integrate ordinary differential equations. In comparison with typical ordinary differential equation solvers, the methods used to integrate partial differential equations are of very low order. Low-order schemes are used for two basic reasons.
First, the approximation of the time derivative is not the only source of finitedifferencing error in the solution of partial differential equations; other errors arise
through the approximation of the spatial derivatives . In many circumstances the
largest errors in the solution are introduced through the numerical evaluation of
the spatial derivatives, so it is pointless to devote additional computational resources to higher-order time-differencing. The second reason for using low-order
methods is that practical limitations on computational resources often leave no
other choice.
2.3.1 The Oscillation Equation: Phase-Speed
and Amplitude Error
Hundreds of papers have been written investigating various techniques for the
finite-difference solution of the advection equation (2.10), many of which are
listed in the extensive review by Rood (1987). The vastness of this body of literature is a testament to the subtle tradeoffs involved in the selection of the "best"
numerical method for even very simple equations. It might be supposed that the
relative accuracy of different methods could be easily determined by comparing
their respective truncation errors. The analysis of truncation error is, however,
most effective at predicting the behavior of well-resolved waves, and the most
serious errors are often found in the poorly resolved waves. The accuracy of both
weil resolved and poorly resolved waves can be better examined by extending the
standard Von Neumann analysis to study the phase-speed and amplitude error in
each Fourier component of the numerical solution.
2. BasicFinite-Difference Methods
spatial derivatives. In contrast, storage limitations dictate that tP be retained at as
few time levels as possible, and the only time levels available are those from previous iterations . Thus , higher-order finite-difference approximations to the time
derivative are inherently one-sided.
The following discussion of the effects of time-differencing on the numerical
solution is transferable, after minor modification, to situations where the spatial
derivatives are approximated by centered differences. In addition, this discussion
provides an exact analysis of the influence of time-differencing on schemes, such
as the spectral method, where finite differences are not used to evaluate the spatial derivatives. Nevertheless, one must be careful not to assume that space and
time differences are completely independent. Indeed, techn iques such as the LaxWendroff method cannot be properly analyzed without understanding the interaction between space truncation error and time truncation error. The combined
effects of space- and time-differencing will be discussed, together with schemes
like the Lax-Wendroff method, in Section 2.5.
Time-differencing formulae used in the numerical solution of partial differential equations are related, naturally enough, to the numerical methods used to
integrate ordinary differential equations. In comparison with typical ordinary differential equation solvers, the methods used to integrate partial differential equations are of very low order. Low-order schemes are used for two basic reasons.
First, the approximation of the time derivative is not the only source of finitedifferencing error in the solution of partial differential equations; other errors arise
through the approximation of the spatial derivatives . In many circumstances the
largest errors in the solution are introduced through the numerical evaluation of
the spatial derivatives, so it is pointless to devote additional computational resources to higher-order time-differencing. The second reason for using low-order
methods is that practical limitations on computational resources often leave no
other choice.
2.3.1 The Oscillation Equation: Phase-Speed
and Amplitude Error
Hundreds of papers have been written investigating various techniques for the
finite-difference solution of the advection equation (2.10), many of which are
listed in the extensive review by Rood (1987). The vastness of this body of literature is a testament to the subtle tradeoffs involved in the selection of the "best"
numerical method for even very simple equations. It might be supposed that the
relative accuracy of different methods could be easily determined by comparing
their respective truncation errors. The analysis of truncation error is, however,
most effective at predicting the behavior of well-resolved waves, and the most
serious errors are often found in the poorly resolved waves. The accuracy of both
weil resolved and poorly resolved waves can be better examined by extending the
standard Von Neumann analysis to study the phase-speed and amplitude error in
each Fourier component of the numerical solution.