A static memory sparse spectral method for time-fractional PDEs
We discuss a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on [0, T] × Ω with Ω ⊂ R d without the excessive memory requirements associated with the nonlocal fractional derivative operator. Our approach combines recent advances in the de...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2023
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| Summary: | We discuss a method which provides accurate numerical solutions to fractional-in-time partial differential equations
posed on [0, T] × Ω with Ω ⊂ 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 α > 0. An attractive feature of the method
is that it has minimal theoretical overhead when using it on any domain Ω ⊂ 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. |
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