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In this paper, we study a fast and linearized finite difference method tosolve the nonlinear time-fractional wave equation with multi fractional orders.We first propose a discretization to the multi-term Caputo derivative based onthe recently established fast L2-1{\sigma} formula and a weighted approach.Then we apply the discretization to construct a fully fast linearized discretescheme for the nonlinear problem under consideration. The nonlinear term, whichjust fulfills the Lipschitz condition, will be evaluated on the previous timelevel. Therefore only linear systems are needed to be solved for obtainingnumerical solutions. The proposed scheme is shown to have second-orderunconditional convergence with respect to the discrete H1-norm. Numericalexamples are provided to justify the efficiency.

A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation