{"action":"created","issue":{"url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/286","labels_url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/286/labels{/name}","comments_url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/286/comments","events_url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/286/events","html_url":"https://github.com/ApproxFun/ApproxFun.jl/issues/286","id":122775227,"number":286,"title":"Add ETDRK4 fourth-order time-stepping for stiff PDEs","user":{"login":"MikaelSlevinsky","id":8176037,"avatar_url":"https://avatars.githubusercontent.com/u/8176037?v=3","gravatar_id":"","url":"https://api.github.com/users/MikaelSlevinsky","html_url":"https://github.com/MikaelSlevinsky","followers_url":"https://api.github.com/users/MikaelSlevinsky/followers","following_url":"https://api.github.com/users/MikaelSlevinsky/following{/other_user}","gists_url":"https://api.github.com/users/MikaelSlevinsky/gists{/gist_id}","starred_url":"https://api.github.com/users/MikaelSlevinsky/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/MikaelSlevinsky/subscriptions","organizations_url":"https://api.github.com/users/MikaelSlevinsky/orgs","repos_url":"https://api.github.com/users/MikaelSlevinsky/repos","events_url":"https://api.github.com/users/MikaelSlevinsky/events{/privacy}","received_events_url":"https://api.github.com/users/MikaelSlevinsky/received_events","type":"User","site_admin":false},"labels":[{"url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/labels/feature","name":"feature","color":"bfdadc"}],"state":"open","locked":false,"assignee":null,"milestone":null,"comments":6,"created_at":"2015-12-17T16:52:51Z","updated_at":"2015-12-18T12:23:20Z","closed_at":null,"body":"The Kassam and Trefethen [classic](https://people.maths.ox.ac.uk/trefethen/publication/PDF/2005_111.pdf) uses a contour integral to stabilize coefficients in the scheme (Eq (2.5)), but I just added more efficient Taylor expansions near the numerical instability.\r\n\r\nThe Taylor expansions require about 18 coefficients in double precision, but the contour integral requires at least 32 calls to `exp` (and uses complex numbers), so this is unequivocally better.\r\n\r\nImplementing the scheme should now be straightforward."},"comment":{"url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/comments/165766718","html_url":"https://github.com/ApproxFun/ApproxFun.jl/issues/286#issuecomment-165766718","issue_url":"https://api.github.com/repos/ApproxFun/ApproxFun.jl/issues/286","id":165766718,"user":{"login":"MikaelSlevinsky","id":8176037,"avatar_url":"https://avatars.githubusercontent.com/u/8176037?v=3","gravatar_id":"","url":"https://api.github.com/users/MikaelSlevinsky","html_url":"https://github.com/MikaelSlevinsky","followers_url":"https://api.github.com/users/MikaelSlevinsky/followers","following_url":"https://api.github.com/users/MikaelSlevinsky/following{/other_user}","gists_url":"https://api.github.com/users/MikaelSlevinsky/gists{/gist_id}","starred_url":"https://api.github.com/users/MikaelSlevinsky/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/MikaelSlevinsky/subscriptions","organizations_url":"https://api.github.com/users/MikaelSlevinsky/orgs","repos_url":"https://api.github.com/users/MikaelSlevinsky/repos","events_url":"https://api.github.com/users/MikaelSlevinsky/events{/privacy}","received_events_url":"https://api.github.com/users/MikaelSlevinsky/received_events","type":"User","site_admin":false},"created_at":"2015-12-18T12:23:20Z","updated_at":"2015-12-18T12:23:20Z","body":"Padé approximants are also defined for `f` satisfying a formal power series."}}