Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021026
Keywords: Finite difference method, whitney forms, differential geometry, differential forms, vector-valued forms, co-vector valued forms, electromagnetism, elasticity
@article{M2AN_2021__55_4_1439_0,
author = {Kettunen, Lauri and Lohi, Jonni and R\"abin\"a, Jukka and M\"onk\"ol\"a, Sanna and Rossi, Tuomo},
title = {Generalized finite difference schemes with higher order {Whitney} forms},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1439--1460},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {4},
doi = {10.1051/m2an/2021026},
mrnumber = {4284398},
zbl = {07523504},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021026/}
}
TY - JOUR AU - Kettunen, Lauri AU - Lohi, Jonni AU - Räbinä, Jukka AU - Mönkölä, Sanna AU - Rossi, Tuomo TI - Generalized finite difference schemes with higher order Whitney forms JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1439 EP - 1460 VL - 55 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021026/ DO - 10.1051/m2an/2021026 LA - en ID - M2AN_2021__55_4_1439_0 ER -
%0 Journal Article %A Kettunen, Lauri %A Lohi, Jonni %A Räbinä, Jukka %A Mönkölä, Sanna %A Rossi, Tuomo %T Generalized finite difference schemes with higher order Whitney forms %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1439-1460 %V 55 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021026/ %R 10.1051/m2an/2021026 %G en %F M2AN_2021__55_4_1439_0
Kettunen, Lauri; Lohi, Jonni; Räbinä, Jukka; Mönkölä, Sanna; Rossi, Tuomo. Generalized finite difference schemes with higher order Whitney forms. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 4, pp. 1439-1460. doi: 10.1051/m2an/2021026
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