Variable-order fractional wave equation: Analysis, numerical approximation, and fast algorithm
Jinhong Jia, Chuanting Jiang, Yiqun Li, Mengmeng Liu, Wenlin Qiu
TL;DR
This work addresses a local modification of a variable-order fractional wave equation to model diffusive waves in viscoelastic media with evolving properties. It develops an equivalent reformulation to prove well-posedness and high-order regularity, introduces a Ritz-Volterra finite element projection to handle the convolution term, and derives rigorous error estimates for fully discrete schemes. A fast divide-and-conquer algorithm leveraging translational invariance reduces the convolution cost from $O(MN^2)$ to $O(MN\log^2 N)$. Numerical experiments confirm the theoretical convergence rates and demonstrate substantial computational efficiency, making high-order, large-scale simulations of variable-order fractional wave propagation feasible.
Abstract
We investigate a local modification of a variable-order fractional wave equation, which describes the propagation of diffusive wave in viscoelastic media with evolving physical property. We incorporate an equivalent formulation to prove the well-posedness of the model as well as its high order regularity estimates. To accommodate the convolution term in the reformulated model, we adopt the Ritz-Volterra finite element projection and then derive the rigorous error estimate for the fully-discretized finite element scheme. To circumvent the high computational cost from the temporal integral term, we exploit the translational invariance of the discrete coefficients associated with the convolution structure and construct a fast divide-and-conquer algorithm which reduces the computational complexity from $O(MN^2)$ to $O(MN\log^2 N)$. Numerical experiments are provided to verify the theoretical results and to demonstrate the accuracy and efficiency of the proposed method.