arXiv subjects

Shun Sato

Publications and source records attributed to Shun Sato.

21 records · Page 2Linked to original sources

arXiv2017

On Spatial Discretization of Evolutionary Differential Equations on the Periodic Domain with a Mixed Derivative

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.

math.NA↗

arXiv2017

Making Bidirected Graphs Strongly Connected

We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time algorithm for finding an optimal solution. For the latter problem, we give a linear-time algorithm for finding a feasible solution whose size is equal to the obvious lower bound or more than that by one.

cs.DM↗

arXiv2016

A novel discrete variational derivative method using "average-difference methods"

We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.

math.NA↗