A fast, spectrally accurate homotopy based numerical method for solving nonlinear differential equations

Andrew C. Cullen, Simon R. Clarke

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9 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)106-118
Number of pages13
JournalJournal of Computational Physics
Publication statusPublished - 15 May 2019


  • Analysis
  • Computational
  • Homotopy
  • Method
  • Nonlinear
  • Numerical

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