The Erfc-Log Filter and the Asymptotics of the Euler and Vandeven Sequence Accelerations

We introduce a new filter or sum acceleration method which is the complementary error function with a logarithmic argument. It was inspired by the large order asymptotics of the Euler and Vandeven accelerations, which we show are both proportional to the erfc function also. We also show the relationship between Vandeven’s filter, the Erfc-Log filter and the “lagged-Euler” method. The theory for the last of these is used to predict a spatially-varying optimal order for filtering of a Fourier or Chebyshev series for a function with a discontinuity, front or shock.