Volume 55, Number 4, July-August 2021
|Page(s)||1439 - 1460|
|Published online||13 July 2021|
Generalized finite difference schemes with higher order Whitney forms
University of Jyväskylä, Faculty of Information Technology, University of Jyväskylä, 35, FI-40014 Jyväskylä, Finland
* Corresponding author: firstname.lastname@example.org
Accepted: 22 May 2021
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.
Mathematics Subject Classification: 35L05 / 35L10 / 58G16 / 58G20 / 58G40
Key words: Finite difference method / whitney forms / differential geometry / differential forms / vector-valued forms / co-vector valued forms / electromagnetism / elasticity
© EDP Sciences, SMAI 2021
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