In astrophysical regimes where the collisional excitation of hydrogen atoms is relevant, the cross-sections for the interactions of hydrogen atoms with electrons and protons are necessary for calculating line profiles and intensities. In particular, at relative velocities exceeding ∼1000 km s^(−1), collisional excitation by protons dominates over that by electrons. Surprisingly, the H–H^+ cross-sections at these velocities do not exist for atomic levels of n ≥ 4, forcing researchers to utilize extrapolation via inaccurate scaling laws. In this study, we present a faster and improved algorithm for computing cross-sections for the H–H^+ collisional system, including excitation and charge transfer to the n≥ 2 levels of the hydrogen atom. We develop a code named bdscx which directly solves the Schrödinger equation with variable (but non-adaptive) resolution and utilizes a hybrid spatial-Fourier grid. Our novel hybrid grid reduces the number of grid points needed from ∼4000n^6 (for a ‘brute force’, Cartesian grid) to ∼2000n4 and speeds up the computation by a factor of ∼50 for calculations going up to n= 4. We present (l, m)-resolved results for charge transfer and excitation final states for n= 2–4 and for projectile energies of 5–80 keV, as well as fitting functions for the cross-sections. The ability to accurately compute H–H+ cross-sections to n= 4 allows us to calculate the Balmer decrement, the ratio of Hα to Hβ line intensities. We find that the Balmer decrement starts to increase beyond its largely constant value of 2–3 below 10 keV, reaching values of 4–5 at 5 keV, thus complicating its use as a diagnostic of dust extinction when fast (∼1000 km s^(−1)) shocks are impinging upon the ambient interstellar medium.