By Robert Vichnevetsky
There was a growing to be curiosity within the use of Fourier research to ascertain questions of accuracy and balance of numerical tools for fixing partial differential equations. this type of research can produce quite beautiful and important effects for hyperbolic equations. This booklet offers necessary reference fabric for these involved in computational fluid dynamics: for physicists and engineers who paintings with pcs within the research of difficulties in such assorted fields as hydraulics, gasoline dynamics, plasma physics, numerical climate prediction, and shipping strategies in engineering, and who have to comprehend the results of the approximations they use; and for utilized mathematicians focused on the extra theoretical points of those computations
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Additional info for Fourier analysis of numerical approximations of hyperbolic equations
297) or (1. 298) is the velocity potential of a double-layer distribution on S or Γ, respectively. The first-kind boundary value problem (prescribing the value of U on S0 [or Γ0 ]) specifies that U t |S0 = 0 and gives rise to the following Fredholm integral equations of the first kind for the unknown single-layer distribution σD , → − → − − → − rs , r ds = U 0 (− σ D r G3 → rs ) , → r s ∈ S0 (1. 299) → − → − − → − rs , r dΓ = U 0 (− σ D r G2 → rs ) , → r s ∈ Γ0 . (1. 300) S0 Γ0 © 2002 by Chapman & Hall/CRC The second-kind boundary value problem specifies that ∂U t ∂n = 0 and S0 (Γ0 ) gives rise to the following Fredholm integral equations of the first kind for the unknown double-layer distribution σN , σN S0 σN Γ0 → − ∂2 ∂U 0 − rs , r ds = G3 → , ∂ns ∂n ∂ns → − → − ∂U 0 ∂2 → r rs , r dΓ = G2 − , ∂ns ∂n ∂ns − → r − → rs ∈ S 0 (1.
1. 304) Imposition of the boundary conditions U t (a + 0, θ) = U t (a − 0, θ) , θ ∈ (0, θ0 ) ∂U t ∂U t = , θ ∈ (θ0 , π) ∂r r=a+0 ∂r r=a−0 © 2002 by Chapman & Hall/CRC (1. 305) (1. 306) leads to the dual series equations for the unknown coefficients ∞ 0 an jn (ka) h(1) n (ka) Pn (cos θ) = −U (a, θ) , θ ∈ (0, θ0 ) (1. 307) n=0 ∞ θ ∈ (θ0 , π) an Pn (cos θ) = 0, (1. 308) n=0 where the incident potential has the expansion in spherical harmonics ∞ U 0 (a, θ) = −eika cos θ = − in jn (ka) Pn (cos θ) .
There are two descriptions of the electromagnetic field radiated by the vertical dipole. The first uses the “generating” functions Hφ and Eφ with the formulae (1. 89)–(1. 90). This approach is restricted to the axially symmetric case (∂/∂φ ≡ 0). 1, the electric Debye potential U and the magnetic Debye potential V . Both potentials satisfy the Helmholtz equation ∇2 U + k 2 U = ∇2 V + k 2 V = 0, (1. 202) and the related electromagnetic fields are given by the formulae (1. 80). (To avoid confusion with the formulae in , note that we use the harmonic time dependence exp (−iωt) instead of exp (+iωt)).
Fourier analysis of numerical approximations of hyperbolic equations by Robert Vichnevetsky