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Additional resources for An Introduction to Atmospheric Modeling
In such a case, the error of the solution usually will not be decreased by reﬁning the grid. This illustrates that if the value of c is such that x0 lies outside of the domain of dependence, it is not possible for the solution of the ﬁnite-difference equation to approach the solution of the differential equation, no matter how ﬁne the mesh becomes. A ﬁnite-difference scheme for which the discretization error can be made small for any initial condition is called a convergent ﬁnite difference scheme.
Later we will extend this list to include additional desirable properties. 10 Summary 41 offs. For example, a more accurate scheme is usually more complicated and expensive than a less accurate scheme. We have to ask whether the additional complexity and computational expense are justiﬁed by the increased accuracy. The answer depends on the particular application. An Introduction to Atmospheric Modeling 42 Basic Concepts Problems 1. Prove that a ﬁnite-difference scheme with errors of order n gives exact derivatives for polynomial functions of degree n or less.
This is an important weakness of the method, because the equations used in numerical models are typically nonlinear and/or have spatially variable coefﬁcients -- if this were not true we would solve them analytically! The point is that von Neumann’s method can sometimes tell us that a scheme is stable, when in fact it is unstable. In such cases, the instability arises from nonlinearity and/or through the effects of spatially variable coefﬁcients. This kind of instability will be discussed in Chapter 6.