Geodesic
A matrix-free Legendre spectral method for initial-boundary value problems
Electronic transactions on numerical analysis, Tome 45 (2016), pp. 283-304
We present a Legendre spectral method for initial-boundary value problems with variable coefficients and of arbitrary dimensionality, where the computational work in each time step scales linearly with the number of unknowns. Boundary conditions are enforced weakly, allowing for stable solutions of many classes of problems. Working in coefficient space, derivatives can be evaluated recursively in linear time. We show how also the action of variable coefficients can be implemented without transforming back to coordinate space using a recursive, linearly scaling matrix-free algorithm, under the assumption that the coefficients vary on a much longer scale than the solution. We also prove that spectral accuracy is preserved for smooth solutions. Numerical results for the wave equation in two and three dimensions corroborate the theoretical predictions.
Classification : 65M12, 65M15, 65M20, 65M70
Keywords: spectral methods, matrix-free methods, method of lines, stability, computational wave propagation, boundary conditions 
@article{ETNA_2016__45__a11,
     author = {Brumm,  Bernd and Kieri,  Emil},
     title = {A matrix-free {Legendre} spectral method for initial-boundary value problems},
     journal = {Electronic transactions on numerical analysis},
     pages = {283--304},
     year = {2016},
     volume = {45},
     zbl = {1352.65387},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2016__45__a11/}
}
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AU  - Brumm,  Bernd
AU  - Kieri,  Emil
TI  - A matrix-free Legendre spectral method for initial-boundary value problems
JO  - Electronic transactions on numerical analysis
PY  - 2016
SP  - 283
EP  - 304
VL  - 45
UR  - http://geodesic.mathdoc.fr/item/ETNA_2016__45__a11/
LA  - en
ID  - ETNA_2016__45__a11
ER  - 
%0 Journal Article
%A Brumm,  Bernd
%A Kieri,  Emil
%T A matrix-free Legendre spectral method for initial-boundary value problems
%J Electronic transactions on numerical analysis
%D 2016
%P 283-304
%V 45
%U http://geodesic.mathdoc.fr/item/ETNA_2016__45__a11/
%G en
%F ETNA_2016__45__a11
Brumm,  Bernd; Kieri,  Emil. A matrix-free Legendre spectral method for initial-boundary value problems. Electronic transactions on numerical analysis, Tome 45 (2016), pp. 283-304. http://geodesic.mathdoc.fr/item/ETNA_2016__45__a11/