Error estimates of compact and hybrid Richardson schemes for the parabolic equation

Hybrid time integration has received more and more attention in recent years due to its inherent computational advantages including all-at-once implementation and without need of multiple start values. In this letter we establish the compact and hybrid Richardson schemes and show that they are unconditionally convergent in theory under the framework of the energy method for the first time to solve the parabolic equation. The numerical example verifies the theoretical results. The analysis procedure outlined can be extended to many other linear/nonlinear evolution equations.