A Fast, Spectrally Accurate Homotopy Based Numerical Method For Solving Nonlinear Differential Equations
| Autor: | Simon R Clarke, Andrew C. Cullen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Numerical Analysis
Physics and Astronomy (miscellaneous) Discretization Applied Mathematics Homotopy Numerical analysis MathematicsofComputing_NUMERICALANALYSIS Numerical Analysis (math.NA) Computer Science Applications Computational Mathematics Nonlinear system Linear differential equation Modeling and Simulation ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION FOS: Mathematics Applied mathematics Boundary value problem Mathematics - Numerical Analysis Homotopy analysis method Mathematics Sparse matrix |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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