By Randall D.
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69). e. the shape of f is determined by the initial condition. Eq. 74) means that u is constant along the line ξ ≡ x – ct = constant . 70), we chose f ≡ F . 69) which satisﬁes the initial condition. We see that an initial value simply “moves along” the lines of constant ξ . The initial “shape” of u ( x ) , namely F ( x ) , is just carried along by the wind. From a physical point of view this is obvious. 69). We construct a grid, as in Fig. 4. 69) is n+1 n n n u j – u j – 1 uj – uj - = 0 . 75) Here we have used the forward difference quotient in time and the backward difference quotient in space.
Also note that the Heun scheme is explicit, and does not require the past history (does An Introduction to Atmospheric Modeling 52 A Survey of Time-Differencing Schemes for the Oscillation and Decay Equations not require l > 0 ). Still, it has second order accuracy, because of the iteration. This illustrates that iteration can increase the order of accuracy. A famous example of an iterative scheme is the fourth-order Runge-Kutta scheme. This is an excellent scheme when f has a simple form, but it is not economically practical when f is complicated.
The upstream scheme under consideration here is, therefore, consistent. 6 Discretization error and convergence There are two sources of error in a numerical solution. One is the round-off error, which is the difference of a numerical solution from the “exact” solution of the ﬁnite n difference equation, u j , and is a property of the machine being used (and to some extent the details of the program). The other source of error is the discretization error deﬁned by An Introduction to Atmospheric Modeling 26 Basic Concepts n u j – u ( j∆x, n∆t ) .
An Introduction to Atmospheric Modeling by Randall D.