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

TL;DR

This paper introduces a computationally efficient numerical method for solving integro-differential equations with increasingly smooth memory kernels, significantly reducing computational costs and applicable to various physical, chemical, and biological systems.

Contribution

The paper proposes a novel numerical ansatz that decreases computational complexity from T^2 to T*C(T) for integro-differential equations with smooth memory kernels, enabling faster simulations.

Findings

Algorithm reduces computational cost from T^2 to T*lnT for certain kernels.

Demonstrated effectiveness on simple equations and spin-boson model.

Applicable to a wide range of systems with smooth memory effects.

Abstract

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 T*C(T), where T is the total simulation time and C(T) is some function. For instance, C(T) is equal to lnT 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. First, we compare the algorithm to a typical numerical procedure for a simple integro-differential equation. Second, we solve the NIBA equations for the spin-boson model in real time.