By Robert Vichnevetsky
There was a growing to be curiosity within the use of Fourier research to check questions of accuracy and balance of numerical tools for fixing partial differential equations. this sort of research can produce rather beautiful and invaluable effects for hyperbolic equations.
This publication presents important reference fabric for these focused on computational fluid dynamics: for physicists and engineers who paintings with desktops within the research of difficulties in such diversified fields as hydraulics, gasoline dynamics, plasma physics, numerical climate prediction, and delivery tactics in engineering, and who have to comprehend the results of the approximations they use; and for utilized mathematicians excited about the extra theoretical elements of those computations.
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Extra info for Fourier Analysis of Numerical Approximations of Hyperbolic Equations
34 CHAPTER 2 FIG. 6. Phase velocity of implicit semi-discretizations of the wave equation U,, = c Uxs. 59) have a truncation error which is O(h2)). 5) is found to be (see Fig. 6) Note the fact that with linear finite elements (J3 =3) we have c*(«)>c for 0 < \a>\ =£ TT/C. 5 By contrast, c*
Velocity error. 16). 11) for the corresponding phase velocity, we readily find where the expansion is to be terminated after the appropriate number of terms. These functions are illustrated in Fig. 2. 3. Limiting case (jSf-»oo). 8) is convergent for a>h\ < TT. In fact, it converges on that interval to ica as K ->oo. 8) over the whole real axis is simply that illustrated in Fig. 3. 4. Relation to the cardinal function. 64). 38 CHAPTER 3 FIG. 1. 10) for increasing values of K. 19) may be expressed simply as HIGHER ORDER SEMI-DISCRETIZATIONS 39 FIG.
When Re A(a>) =sO, and the < sign holds for some oj, then, in general, \\un\\\ decreases with time. Dissipation (of energy) is introduced spuriously by the approximation. Spurious dispersion: Exact solutions of the advection equation are nondispersive. That is, the shape of the initial function is preserved: The nonconstancy of c* in discrete approximations results in a nonconstant velocity of propagation of the harmonic components into which a nonsinusoidal initial function is resolved by Fourier analysis.