Lattice points on circles and discrete velocity models for the Boltzmann equation
2006 (English)In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 37, no 6, 1903-1922 p.Article in journal (Refereed) Published
The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].
Place, publisher, year, edition, pages
2006. Vol. 37, no 6, 1903-1922 p.
Boltzmann equation, discrete velocity model, multiplicative functions, distribution of Gaussian primes, half-integral weight, invariants, forms
IdentifiersURN: urn:nbn:se:kth:diva-15604DOI: 10.1137/040618916ISI: 000236805700010ScopusID: 2-s2.0-33750971724OAI: oai:DiVA.org:kth-15604DiVA: diva2:333646
QC 201005252010-08-052010-08-052016-08-10Bibliographically approved