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

## TL;DR

This paper introduces a fast, linearized finite difference method for solving nonlinear multi-term time-fractional wave equations, achieving second-order convergence and computational efficiency.

## Contribution

It develops a novel discretization for multi-term Caputo derivatives and constructs a fully linearized scheme that simplifies solving nonlinear problems.

## Key findings

- Second-order convergence in discrete H1-norm.
- Efficient solution of nonlinear multi-term fractional wave equations.
- Numerical results confirm the method's accuracy and efficiency.

## Abstract

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