  Starting with the asymptotic expansion of the error equation of the shifted
Gr\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted
Gr\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction
terms. We then apply one special case of the modified WSGL formula to solve
multi-term fractional ordinary and partial differential equations, and we prove
the linear stability and second-order convergence for both smooth and
non-smooth solutions. We show theoretically and numerically that numerical
solutions up to certain accuracy can be obtained with only a few correction
terms. Moreover, the correction terms can be tuned according to the fractional
derivative orders without explicitly knowing the analytical solutions.
Numerical simulations verify the theoretical results and demonstrate that the
new formula leads to better performance compared to other known numerical
approximations with similar resolution.
