We propose digital correction methods for recovering aberrated spectra in a portable low-cost miniaturized grating-based spectrometer, which is modelled in optical and numerical simulations. To realize the digital spectrum recovery, different Point Spread Function (PSF) modelling approaches for wavelength-dependent PSFs are proposed and implemented into four digital correction algorithms, namely inverse Fourier Transform (IFT) deconvolution, Wiener deconvolution, Lucy-Richardson (LR) and Landweber iterative algorithm. Results show that both LR and Landweber algorithms can improve the spectral resolution by about a factor of two. The enhanced spectral resolution is comparable to that of commercial table-top spectrometers, while our spectrometer has a much smaller packaging volume of about one cubic inch.