English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The subject of this paper is the design of efficient and stable spectralmethods for time-dependent partial differential equations in unit balls. Wecommence by sketching the desired features of a spectral method, which isdefined by a choice of an orthonormal basis acting in the spatial domain. Wecontinue by considering in detail the choice of a $W$-function basis in a discin $\mathbb{R}^2$. This is a nontrivial issue because of a clash between twoobjectives: skew symmetry of the differentiation matrix (which ensures interalia that the method is stable) and the correct behaviour at the origin. Weresolve it by representing the underlying space as an affine space andsplitting the underlying functions. This is generalised to any dimension $d\geq 2$ in a natural manner and the paper is concluded with numerical examplesthat demonstrate how our choice of basis attains the best outcome out of anumber of alternatives.

A framework for stable spectral methods in $d$-dimensional unit balls