Enhanced rational multi-derivative integrator for singular problems with application to advection equations

Singular problems with discontinuities or steep gradients challenge traditional numerical integration methods. This study introduces the Rational Multi-Derivative Integrator (RMDI), designed specifically for such problems in ordinary and partial differential equations. The RMDI employs rational approximations and neural data to effectively manage singularities, enhancing accuracy and stability. Benchmark tests on singular ODEs and Advection equations show the RMDI reduces error rates by up to 99% and converges at singular points with errors within 1.5% of exact solutions. In real-world applications, the RMDI consistently outperforms existing methods, confirming its potential for practical use in scientific and engineering fields.