By Randall D.
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J Here the summation obviously must be taken over a ﬁnite number of grid points. This is not an important limitation because in practice we are always dealing with a ﬁnite n number of grid points. If the sum is bounded, then each u j must also be bounded. n+1 Whereas in the direct method we checked max ( j ) u j check n 2 ∑ ( uj ) , here in the energy method we . The two approaches are therefore somewhat similar. 91) . j For simplicity, suppose that u is periodic in x , and consider a summation over one complete cycle.
82) 28 n+1 n n n n Basic Concepts n u j – u j – 1 uj + 1 – uj – uj uj - + c - ---------------------- = 0 . 83) This form avoids the use of if-tests and is also convenient for use in pencil-and-paper analysis, discussed later. In summary: Truncation error measures the accuracy of an approximation to a differential operator or operators. It is a measure of the accuracy with which a differential equation has been approximated. Discretization error measures the accuracy with which the solution of the differential equation has been approximated.
87) For 0 ≤ µ ≤ 1 we have interpolation. For µ < 0 or µ > 1 we have extrapolation. Note that for the case of interpolation, u jn + 1 will be intermediate in value between u jn– 1 and u jn . For instance, if u jn– 1 and u jn are both ≥ 0 , then u jn will also be ≥ 0 . 6: Diagram illustrating the concepts of interpolation and extrapolation. See text for details. extrapolation, u jn + 1 will lie outside the range of u jn– 1 and u jn . We use both interpolation and extrapolation extensively in this course.
An Introduction to Atmospheric Modeling by Randall D.