A fast, spectrally accurate homotopy based numerical method for solving nonlinear differential equations
Introduction
The Homotopy Analysis Method (HAM) [17] is a modern technique for iteratively constructing analytic solutions to nonlinear equations, that can be considered as an extension of the Lyapunov method [20]. Applying this scheme to vexatious nonlinear problems has demonstrated that the HAM has advantageous convergence properties, relative to those exhibited by other techniques [18], [12], [13]. The technique works by decomposing a nonlinear problem into a sequentially coupled series of linear differential equations, through the use of a homotopy, which is a concept rooted in differential geometry. However, as an analytic method, it relies on being able to solve the linear equations either by hand, or through the use of computer algebra packages. This requirement limits the scope of problems that can be approached using the HAM. As such, there is particular interest in developing new numerical schemes, that leverage the convergence properties of the HAM. As a consequence of the method's convergence properties, a particular focus has been placed upon high-order accurate methods, which can produce fundamentally more accurate and reliable solutions for numerical domains discretized at lower resolutions, which in turn impacts upon the computational cost of these schemes.
These high-order accurate methods can in turn be broadly categorised into local and global (or spectral) methods, with the later involving discretisations of derivatives that depend upon all the points in the numerical domain, while local methods calculate derivatives at each point in the domain in terms of its adjacent elements [30]. Local high-order methods are employed in multi-block finite difference methods, some finite volume methods, and stabilised finite element methods; whereas Discontinuous Galerkin (DG), DG spectral element methods, and spectral volume and difference methods all fall under the categorisation of global methods [4], [32], [36]. These schemas are typically considered as the basis for solving unsteady problems upon a wide range of domains. However, while these schemes can be applied to solve steady nonlinear problems by coupling them within an iterative framework, the schemes can introduce significant computational costs. Furthermore, these schemes are not guaranteed to remain stable for sufficiently long to resolve the steady solution.
While a number of specific schemes to approach steady, nonlinear problems have been proposed, including Newton iteration, AiTEM [35], nonlinear Conjugate Gradient [7] and Multigrid based methods [10], [27], Newton iteration remains amongst the most commonly employed. These schemes all can be considered as linear solvers that are iterated upon in order to converge upon the motivating nonlinear problem. However, as was the case with the earlier examples, the iterative schemes that result often scale poorly with the grid resolution.
This paper we will outline a new approach to constructing numerical solutions through the HAM, called the Gegenbauer Homotopy Analysis Method (GHAM), and examine its numerical performance relative to other HAM based approaches, and other numerical schemes for solving nonlinear differential equations. We first describe the HAM in Section 2, in the context of its original formulation as a semi-analytic scheme, and as a numerical scheme when solved using the Spectral Homotopy Analysis Method (SHAM). In Section 3, we introduce the GHAM, and discuss its advantages over previously developed nonlinear solvers. Section 4 will present a theoretical justification of the difference in numerical performance between the GHAM and other techniques. Finally, in Section 5 we show numerical results, convergence, and scaling properties for a fourth-order nonlinear variable coefficient boundary value problem, and compare these results for the Gegenbauer- and Spectral-Homotopy Analysis methods with Newton Iteration and MATLAB's inbuilt boundary value problem routine ‘BVP4C’. These comparisons will allow us to understand the potential for the GHAM to significantly reduce the computational cost of solving steady, nonlinear boundary value problems.
Section snippets
Homotopy Analysis Method
The Homotopy Analysis Method (HAM) is a technique for solving nonlinear equations by constructing a homotopy that defines a smooth, continuous deformation from one equation onto another. If a homotopy can be found between the two equations, it follows that the same homotopy should also hold for the solutions of the two equations, and thus allowing one known solution to an equation to be deformed onto the unknown solution of another equation. Within the context of the HAM, when searching for a
Gegenbauer Homotopy Analysis Method
For all their advantageous numerical properties, Chebyshev polynomials have limited utility for constructing matrix operators for variable coefficient linear boundary value problems, as the resulting matrix operators rapidly become singular with increases in the grid resolution. An alternative approach is to turn to the Gegenbauer (otherwise known as Ultraspherical) polynomials, of which the Chebyshev polynomials are a subset. These Gegenbauer polynomials exhibit the previously introduced
Theoretical scaling
The difference between the GHAM and other comparable numerical schemes can be considered by examining the cost of solving the matrix equations that these schemes produce. Typically, nonlinear solvers require the construction of a new dense matrix inverse at each step of the iterative process, however the Spectral- and Gegenbauer Homotopy Analysis Methods can be considered in terms of a single matrix operator, that is either dense in the case of SHAM, or sparse for GHAM. Thus, a solution process
Numerical testing of the GHAM
To test the validity of the above theoretical scaling properties, we turn to the problem of a two-dimensional flow of a laminar, viscous, incompressible fluid confined within a rectangular domain bounded by moving porous walls, which can be recast as a nonlinear, variable coefficient boundary value problem. This problem has applications to mixing processes, as well as for boundary layer control systems, and was chosen as it has previously been studied by Motsa [23] through the use of SHAM. The
Conclusion
We have presented a new, spectrally accurate scheme for solving steady, nonlinear boundary value problems, that uses the Homotopy Analysis Method to convert the original problem into an infinite sequence of linear differential equations, which are in turn solved upon a Gegenbauer discretisation. Unlike other numerical approaches for steady, nonlinear boundary value problems, the GHAM exhibits quasi-linear growth in time with respect to the grid resolution, and the number of iterations, due to
Acknowledgements
We thank the reviewers for their suggestions, which have clarified important points in this paper.
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