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A Fast, Spectrally Accurate Homotopy Based Numerical Method For Solving Nonlinear Differential Equations
- Publication Year :
- 2018
-
Abstract
- We present an algorithm for constructing numerical solutions to one--dimensional nonlinear, variable coefficient boundary value problems. This scheme is based upon applying the Homotopy Analysis Method (HAM) to decompose a nonlinear differential equation into a series of linear differential equations that can be solved using a sparse, spectrally accurate Gegenbauer discretisation. Uniquely for nonlinear methods, our scheme involves constructing a single, sparse matrix operator that is repeatedly solved in order to solve the full nonlinear problem. As such, the resulting scheme scales quasi-linearly with respect to the grid resolution. We demonstrate the accuracy, and computational scaling of this method by examining a fourth-order nonlinear variable coefficient boundary value problem by comparing the scheme to Newton-Iteration and the Spectral Homotopy Analysis Method, which is the most commonly used implementation of the HAM.
- Subjects :
- Mathematics - Numerical Analysis
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1811.00676
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jcp.2019.01.057