Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

Starting with the asymptotic expansion of the error equation of the shifted Grünwald-Letnikov formula, we derive a new modified weighted shifted Grünwald-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 when the regularity of the solutions is known. We show theoretically and numerically that numerical solutions with good 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 WSGL formula leads to better performance compared to other known numerical approximations with similar resolution.