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 TBn(y) and SBn(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.