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

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 σ 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 H 1 -norm. Numerical examples are provided to justify the efficiency.

Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

“…Two temporal second-order schemes are derived and analyzed for the time multi-term fractional diffusion-wave equation based on the order reduction technique in Reference [22]. A fast and linearized finite difference method to solve a nonlinear time-fractional wave equation with multi fractional orders is studied in Reference [23]. In all these articles it is usually assumed that the solution u(t) (ignoring the space variable) lies in C 2 ([0, T]) or C 3 ([0, T]), but it is well-known that ∂ tt u blows up as t → 0 + in some practical situations [16,24,25].…”

A high-order and fast scheme with variable time steps for the time-fractional Black-Scholes equation

“…As the experimental merit of the SOE technique has been well demonstrated in many previous works, e.g. [6,18,19,28], we mainly focus on the accuracy verification, one may refer to [19,28] for the computational advantage of the fast Alikhanov approximation comparing to the classical approximation while solving the time-fractional partial differential equations. In our computations, the special domain is divided uniformly into M subintervals and the time interval is divided by a general nonuniform grid with N parts.…”

Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients

“…Moreover, we will always employ the sum-of-exponentials (SOE) technique [7] to the proposed schemes while discretizing the Caputo derivative to save the memory and computational costs, since the SOE method does not bring any additional essential differences to the numerical analysis of the nonuniform schemes. One may refer to [16,Section 5.1] for the details of the fast Alikhanov formula and refer to [7,19] for the advantage of the SOE approximation in the computational aspect. The absolute tolerance error ǫ and the cut-off time ∆t of the fast Alikhanov formula (see [16,Lemma 5.1]) are set as ǫ = 10 −12 and ∆t = τ 1 in all of the following tests.…”