Numerical ansatz for solving integro-differential equations with increasingly smooth memory kernels: spin-boson model and beyond

We present an efficient and stable numerical ansatz for solving a class of integro-differential equations. We define the class as integro-differential equations with increasingly smooth memory kernels. The resulting algorithm reduces the computational cost from the usual T^2 to TC(T), where T is the total simulation time and C(T) is some function. For instance, C(T ) is equal to ln T for polynomially decaying memory kernels. Due to the common occurrence of increasingly smooth memory kernels in physical, chemical and biological systems, the algorithm can be applied in quite a wide variety of situations. We demonstrate the performance of the algorithm by examining two cases. Firstly, we compare the algorithm to a typical numerical procedure for a simple integro-differential equation. Secondly, we solve the non-interacting blip approximation equations for the spin-boson model in real time.