Numerical noise suppression for wave propagation with finite elements in first-order form by an extended source term
R. Shamasundar (TU Delft - Applied Geophysics and Petrophysics)
W. Mulder (TU Delft - Applied Geophysics and Petrophysics, Shell Global Solutions International B.V.)
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Abstract
Finite elements can, in some cases, outperform finite-difference methods for modelling wave propagation in complex geological models with topography. In the weak form of the finiteelement method, the delta function is a natural way to represent a point source. If, instead of the usual second-order form, the first-order form of the wave equation is considered, this is no longer true. Fourier analysis for a simple case shows that the spatial operator corresponding to
the first-order form has short-wavelength null-vectors. Once excited, these modes are not seen by the spatial operator but only by the time- stepping scheme and show up as noise. A sourcewith a larger spatial extent, for instance a Gaussian or a tapered sinc, can avoid the excitation of problematic short wavelengths. A series of numerical experiments on a 2-D problem with an exact solution provides a suggestion for the best choice of parameters for these source
term distributions. The tapered sinc provided the best results and the resulting accuracy can be better than that of the second-order form. The higher operation count of the former, however, does not make it more efficient in terms of accuracy for a given computational effort, at least not for the 2-D examples considered here