Fast Summation by Interval Clustering for an Evolution Equation with Memory

2012/01/01 by William McLean · 6 citations
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #Fractional Differential Equations Solutions #Model Reduction and Neural Networks

paper · doi:10.1137/120870505

openalex publication_date 2012/01/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30

Abstract

We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in space. If there are N time levels and M spatial degrees of freedom, then a direct implementation of this method requires O(N²M) operations and O(NM) active memory locations, owing to the presence of a memory term: at each time step, the discrete evolution equation involves a sum over all previous time levels. We show how the computational cost can be reduced to O(MNlog N) operations and O(Mlog N) active memory locations.

Citations

• A Fast Time Stepping Method for Evaluating Fractional Integrals

Cited by

• Development of numerical methods for nonlinear hybrid stochastic functional differential equations with infinite delay
• An accurate spectral method for Maxwell equations in Cole-Cole dispersive media
• A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations
• High order fast algorithm for the Caputo fractional derivative
• Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations
• An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion

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