M4: Adaptive spectral methods in 1D and 2D

Background

Ordinary and partial differential equations (ODEs and PDEs) are used to model a vast number of physical processes that arise throughout scientific disciplines. Typically, such equations cannot be solved analytically, and so are solved numerically. Whilst there are a number of methods for solving differential equations that fit this description, amongst the fastest and most accurate are spectral methods. However, for troublesome functions that are not well-represented by polynomials, the performance of this approach can be poor. Our goal is to recover the impressive convergence for these more challenging problems by using adaptive spectral methods.

Techniques and Challenges

By building on recent work [1,2], the project’s goal is to make adaptive spectral methods into a practical, high-accuracy tool for general scientific computing. We will combine four ideas: Padé approximation to locate nearby singularities, Schwarz-Christoffel conformal maps to enable geometric convergence, barycentric rational interpolants to exploit the irregular grid, and very high order ODE solvers for the time discretisation, as well as the Chebfun/chebop system [3]. There are four aims: develop new data-driven conformal maps as the basis of these methods, investigate links with the adaptive technology of the Chebfun/chebop system, make a truly general-purpose adaptive spectral solver in one-dimension, and explore generalisations to two-dimensions.

Results

Investigating techniques for automatic domain decomposition led us to reconsider how boundary conditions are implemented in one- and higher-dimensional spectral methods. The conventional approach is somewhat heuristic and involves replacing rows of the matrix discretised representing the differential operator with others which enforce the boundary conditions. We have developed a new approach [4], which involves projecting the operator onto a lower degree subspace, which is then completed using the boundary conditions. This technique can be far more generally applied.

The Future

Our current research goals involve progressing research into adaptive spectral domain decomposition, improved conditioning in spectral methods (either through automatic integral reformulations, or new stable coefficient-based methods), investigating fractional-order differential equations [11/42], and using spectral methods for differential eigenvalue and singular value computations.

References

[12/22] Hale N., Trefethen L.N.: Chebfun and numerical quadrature

[11/42] Burrage K., Hale N., Kay D.: An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

[1] Tee T.W., Trefethen L.N.: A Rational Spectral Collocation Method with Adaptively Transformed Chebyshev Grid Points, SIAM J. Sci. Comput. 28, pp. 1798-1811, 2006

[2] Hale N., Tee T.W.: Conformal Maps to Multiply Slit Domains and Applications, SIAM J. Sci. Comput. 31, pp. 3195-3215, 2009

[3] Trefethen L.N., Hale N., Driscoll T., et al: Chebfun Version 4

[4] Hale N., Driscoll T.: Resampling methods for boundary conditions in spectral collocation, (In preparation)