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A static memory sparse spectral method for time-fractional PDEs
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DOI：10.1016/j.jcp.2023.112522.png)
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Abstract
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We discuss a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on [0,T]x Omega with Omega subset of R-d without the excessive memory requirements associated with the nonlocal fractional derivative operator. Our approach combines recent advances in the development and utilization of multivariate sparse spectral methods as well as fast methods for the computation of Gauss quadrature nodes with recursive non-classical methods for the Caputo fractional derivative of general fractional order alpha > 0. An attractive feature of the method is that it has minimal theoretical overhead when using it on any domain Omega subset of R-d on which an orthogonal polynomial basis is available. We discuss the memory requirements of the method, present several numerical experiments demonstrating the method's performance in solving time-fractional PDEs on intervals, triangles and disks and derive error bounds which suggest sensible convergence strategies. As an important model problem for this approach we consider a type of wave equation with time-fractional dampening related to acoustic waves in viscoelastic media with applications in the physics of medical ultrasound and outline future research steps required to use such methods for the reverse problem of image reconstruction from sensor data.
Keywords:
Time-fractional
Caputo derivative
Fractional derivative
Fractional PDE
History-free
Sparse
Banded
Spectral method
Orthogonal polynomials
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