2025-07-24 –, Main Room 3
Radau methods are highly efficient ways to solve stiff ordinary differential equations, yet the state-of-the-art remained Fortran scripts from the 1990s for decades. In this talk, I describe several new innovations made to Radau methods in Julia's OrdinaryDiffEq.jl interface that enhance their performance. Prior knowledge of differential equations is not necessary!
Radau methods are known to be state-of-the-art for the high-accuracy solution of highly stiff ordinary differential equations (ODEs). However, the traditional implementation was specialized to a specific ranges of precision and only derived in floating point, thus limiting the algorithm's ability to be truly general purpose for highly accurate scenarios. To alleviate these constraints, we implemented a fully adaptive Radau method that can derive the coefficients for the Radau method on the fly to any precision. Additionally, the Julia-based implementation includes many modernizations to improve performance, including improved linear algebra integrations and parallelization. In a head-to-head benchmark against the classic Fortran implementation, we demonstrate our implementation is approximately 2x faster across a range of stiff ODEs. We further benchmark our algorithm against several well-reputed numerical integrators for stiff ODEs and find state-of-the-art performance on several test problems, with a 1.5x speed-up over common numerical integrators for stiff ODEs when high precision is required. In this talk, we describe the overall scheme of numerical integrators before diving into some of the intricacies of Radau methods, including the beauty of utilizing of the complex plane to accelerate computations and the adaptive time-stepping and order-selection scheme. Prior experience with numerical integration or differential equation is not required! We will start from basic ideas about differential equations and work our way up.
