  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 mathematical and
numerical analysis for this model is not available in the literature, partly
because the variable-exponent Abel kernel may not be positive definite or
monotonic. We overcome these difficulties to design two numerical 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.
