| 研究実績の概要 |
Firstly, the modified formal Lagrangian structure was applied to practical problems. In particular, we extensively studied various numerical solutions of the viscous Burgers' equation, including shock waves. These show improvement of error in the aspect of conservation law-preserving property. Further applications will be conducted, for instance, novel numerical methods for Euler and Navier-Stokes equations.
At the same time, we further investigated the application of discrete moving frames for constructing symmetry-preserving (and hence conservation law-preserving from Noether's theorem) numerical methods for mechanical systems. Following our previous results on Euler's elastica that preserve SE(3) symmetries, we are currently finalizing a similar work towards SE(3)-preserving numerical methods for the Kepler problem. This approach allows us to preserve the total energy and symplectic structures simultaneously, although the step size may vary.
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