L1 Schemes for time-fractional differential equations: A brief survey and new development

Abstract

We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method. Mathematics Subject Classification (2020) 26A33 · 65M06 · 65M12 · 65M15 · 35R11