Numerical Geometric Integration
Introduction to structure-preserving integrators
Mathematical models of the evolution in time of dynamical systems (whether in biology, economics, or computer animation) generally involve systems of differential equations. The equations often have geometric properties such as symmetry or invariance that a naive numerical method may not respect.
This class will present the main concepts behind geometrically derived integrators designed to properly capture these continuous properties. These schemes include variational, symplectic and symmetric methods, all of which can exhibit dramatically improved numerics even over long time periods. By the end of the class, the students will be expected to have a working knowledge of existing geometric integrators, as well as the tools required to extend them to other fields.
No prior knowledge in numerical integration or dynamical systems will be assumed, and an emphasis will be placed on practical application, with examples drawn from physics, astronomy, graphics, and biology. Homework assignments will involve both theory and implementation.
9 units (3-3-3); second term.