Numerical analysis of high-order methods for variable-exponent fractional diffusion-wave equation

TL;DR

This paper develops and analyzes high-order numerical schemes for a complex variable-exponent fractional diffusion-wave equation, addressing stability and error estimates despite challenges posed by indefinite kernels, with numerical results confirming theoretical predictions.

Contribution

It introduces two high-order schemes for the variable-exponent fractional diffusion-wave equation and provides stability and error analysis despite indefinite convolution kernels.

Findings

Schemes achieve $oldsymbol{ ext{α(0)}}$-order and second-order accuracy in time.

Numerical experiments confirm theoretical stability and error estimates.

Transition behavior of mechanical waves modeled by the equation is demonstrated.

Abstract

This work considers the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Rigorous numerical analysis for this model is not available in the literature, partly because the variable-exponent Abel kernel in the leading term may not be positive definite or monotonic. We adopt the idea of model reformulation to obtain a more tractable form, which, however, still involves an ``indefinite-sign, nonpositive-definite, nonmonotonic'' convolution kernel that introduces difficulties in numerical analysis. We address this issue to design two high-order schemes and derive their stability and error estimate based on the proved solution regularity, with $\alpha(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings and to show the transition behavior of the mechanical (diffusive) wave modeled by variable exponent.