A Fast, Spectrally Accurate Homotopy Based Numerical Method For Solving Nonlinear Differential Equations
For researchers in scientific computing, this method offers a faster and more accurate alternative for solving nonlinear boundary value problems, though it is an incremental improvement over existing HAM implementations.
The paper presents a numerical method for solving 1D nonlinear differential equations that combines the Homotopy Analysis Method with a sparse, spectrally accurate Gegenbauer discretization, achieving quasi-linear scaling with grid resolution. It demonstrates superior accuracy and computational efficiency compared to Newton-Iteration and the standard Spectral Homotopy Analysis Method on a fourth-order nonlinear problem.
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.