Hybrid Chebyshev function bases for sparse spectral methods in parity-mixed PDEs on an infinite domain

We present a numerical spectral method to solve systems of differential equations on an infinite interval y ∈ (- ∞ , ∞) in presence of linear differential operators of the form Q (y)^{(∂ /∂_y) b} (where Q (y) is a rational fraction and b a positive integer). Even when these operators are not parity-preserving, we demonstrate how a mixed expansion in interleaved Chebyshev rational functions TB_n (y) and SB_n (y) preserves the sparsity of their discretization. This paves the way for fast O (Nln ⁡ N) and spectrally accurate mixed implicit-explicit time-marching of sets of linear and nonlinear equations in unbounded geometries.