Title: Semigroup Approximation for Parabolic PDEs Using Operator Splitting
Abstract: We develop a method for numerically solving parabolic partial differential equations (PDEs) by constructing an asymptotic expansion of the associated semigroup, accurate up to third order in time. Employing operator splitting, we decompose the PDE into diffusion and transport-decay components, which are individually exponentiated. We describe the semigroup approximations, provide error analysis, and explain why the method preserves the necessary mathematical properties. Numerical implementation strategies are discussed, highlighting the method's advantages in computational efficiency and accuracy when applied within spectral methods.