2023 Fiscal Year Annual Research Report
Multisymplectic Geometry and Geometric Numerical Integrator for Variational Problems
| Project/Area Number |
20K14365
|
|---|
| Research Institution | Keio University |
|---|
Principal Investigator |
彭 林玉 慶應義塾大学, 理工学部(矢上), 准教授 (90725780)
|
|---|
| Project Period (FY) |
2020-04-01 – 2024-03-31
|
| Keywords | Moving frame / Geometric integration / Symmetry-preserving / Burgers' equation / Kepler problem / Symplectic structure / Euclidean group |
|---|
| Outline of Annual Research Achievements |
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.
|
