SDG 7th biweekly report

Review of the past two weeks

In the last two weeks, my work can be summarized as two parts: Implementing the basic structure of COLDAE from scratch and maintaining the BoundaryValueDiffEq.jl package.

1. With the previously translated COLDAE solver, it is easy to see the structure of COLDAE and the collocation equations, during the past two weeks, I utilized the collocation equations setup subroutine and solution approximating subroutine to construct a simple version of COLDAE solver from scratch, for example, the similar routine like what we did in MIKR methods, construct nonlinear problems using collocation equations and solve the nonlinear system. By testing on the example test problem, the discrete solution is converging now.

2. Fixed the existing issues with BoundaryValueDiffEq.jl, such as Fix benchmarks CI, Fix non-adaptive interpolation, Fix StaticArrays support for MIRK methods and ongoing Fix support for BigFloat u0.

3. The Lobatto and Radau methods for BVP are almost done and CIs are passing now, still need the BigFloat support to be configured to push this forward.

More TODOs:

1. Continue the polishing of the simple COLDAE implementation and test the solver on more example problems to see if there are any bugs or improvements needed. Add the error estimate part and mesh refinement part as well to support more features.

2. Continuing the BigFloat support and getting the Lobatto and Radau solvers PR get merged as soon as possible.

3. Fix the interpolations for MIRK methods, related issues are MIRK4 interpolation bugs.