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

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.