Journal article

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

Andrew C Cullen, Simon R Clarke

Journal of Computational Physics | ACADEMIC PRESS INC ELSEVIER SCIENCE | Published : 2019


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 ..

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