Gabor wave packet applied to solving ODEs
By: Alexander Pletzer, CPPG, PPPL
Recent progress on the utilization of wave packets (Gabor functions) to solve ordinary differential equations (ODEs) arising in wave propagation problems is presented. While many RF codes such as Mets rely on FFT to solve the wave equation in a plasma medium, the need arose to explore numerical schemes that are able to capture more efficiently both small and large scale features. Mode conversion phenomena, for instance, between fast magnetosonic and slow, ion Bernstein waves typically require the resolution of waves with dramatically different wavelengths.
Gabor functions, which are Gaussians with a sine modulation, have been applied in the past 50 years or so to signal analysis and synthesis. However, there appears to be little evidence in the literature about the use of Gabors to solve ODEs. This is surprising as it will be shown that the Galerkin method can be straightforwardly adapted to handle Gabor basis functions instead of the usual hat or Hermite basis functions that characterize the finite element method (FEM). The Gabor element method (GEM) is therefore similar to FEM except that it can solve ODEs of, in principle, arbitrary high order since the Gabors are continuous to all orders. Unlike FFT, GEM produces sparse matrices and allows non-periodic boundary conditions to be applied. To validate GEM, two model problems are considered: a second order Airy equation with a linear turning point (cut-off) and a 4th order Wasov equation describing the coupling of modes with well separated wavelengths.Gabor wave packet applied to solving ODEs