26
1. Introduction
relative accuracy ofthe anelastic and pseudo-incompressible approximations cannot be judged solelyon the basis of their dispersion relations . Nance and Durran (1994) and Nance (1997) compared the accuracy of several different systems
of filtered equations and found that the pseudo-incornpressible system and the
anelastic system suggested by Lipps and Hemler are the most accurate, and that
the anelastic system performs slightly better in the hydrostatic limit, whereas the
pseudo-incornpressible system gives slightly better accuracy when the flow is not
hydrostatic.
1.3 Strategies for Numerical Approximation
A wide variety of different methods have been employed to obtain numerical solutions to the systems of partial differential equations discussed in the preceding
sections of this chapter. Before delving into the details of these methods, we conclude this introductory chapter by comparing some of the most general properties
of the various methods, including the manner in which each method approximates
the value of the unknown function and estimates its derivatives. We will also consider some of the fundamental differences between the numerical algorithms used
to solve elliptic and hyperbolic partial differential equations.
1.3.1 Approximating Calculus with Algebra
Digital computers are not designed to solve differential equations directly. AIthough the digital computer can perform algebraic operations such as addition
and multiplication, it does not have any intrinsic ability to differentiate and integrate functions . As a consequence, every numerical method is designed to convert
the original differential equation into a set of solvable algebraic equations . As part
of this task the continuous functions associated with the original problem must be
represented by a finite set of numbers that can be stored in a computer's memory
or on disk. There are therefore two basic problems that must be addressed by every numerical scheme: how to represent the solution by a finite data set and how
to compute derivatives. There are also two basic solution strategies: grid-point
methods and series expansion methods.
In grid-point methods, each function is described by its value at a set of discrete grid points. Figure l.l shows how j(x), a periodic function on the interval
[0, 27T l, might be represented by its exact value at five different points along the
x-axis. The spacing ofthe grid points can be chosen arbitrarily, although any variations in the grid spacingwill affect the accuracy of the approximation. If apriori
knowledge of the function 's periodicity is available, a natural choice for the five
pieces ofinformation would be (f(27T /5), j(47T/5), ... , j(27T». No assumption
is made about the value of the approximate solution between the points on the numerical mesh. These methods are usually calledfinite-difference methods because
1. Introduction
relative accuracy ofthe anelastic and pseudo-incompressible approximations cannot be judged solelyon the basis of their dispersion relations . Nance and Durran (1994) and Nance (1997) compared the accuracy of several different systems
of filtered equations and found that the pseudo-incornpressible system and the
anelastic system suggested by Lipps and Hemler are the most accurate, and that
the anelastic system performs slightly better in the hydrostatic limit, whereas the
pseudo-incornpressible system gives slightly better accuracy when the flow is not
hydrostatic.
1.3 Strategies for Numerical Approximation
A wide variety of different methods have been employed to obtain numerical solutions to the systems of partial differential equations discussed in the preceding
sections of this chapter. Before delving into the details of these methods, we conclude this introductory chapter by comparing some of the most general properties
of the various methods, including the manner in which each method approximates
the value of the unknown function and estimates its derivatives. We will also consider some of the fundamental differences between the numerical algorithms used
to solve elliptic and hyperbolic partial differential equations.
1.3.1 Approximating Calculus with Algebra
Digital computers are not designed to solve differential equations directly. AIthough the digital computer can perform algebraic operations such as addition
and multiplication, it does not have any intrinsic ability to differentiate and integrate functions . As a consequence, every numerical method is designed to convert
the original differential equation into a set of solvable algebraic equations . As part
of this task the continuous functions associated with the original problem must be
represented by a finite set of numbers that can be stored in a computer's memory
or on disk. There are therefore two basic problems that must be addressed by every numerical scheme: how to represent the solution by a finite data set and how
to compute derivatives. There are also two basic solution strategies: grid-point
methods and series expansion methods.
In grid-point methods, each function is described by its value at a set of discrete grid points. Figure l.l shows how j(x), a periodic function on the interval
[0, 27T l, might be represented by its exact value at five different points along the
x-axis. The spacing ofthe grid points can be chosen arbitrarily, although any variations in the grid spacingwill affect the accuracy of the approximation. If apriori
knowledge of the function 's periodicity is available, a natural choice for the five
pieces ofinformation would be (f(27T /5), j(47T/5), ... , j(27T». No assumption
is made about the value of the approximate solution between the points on the numerical mesh. These methods are usually calledfinite-difference methods because