Benchmarking Numerical Methods for Lattice Equations with the Toda Lattice

TL;DR

This paper evaluates and compares the performance of various numerical methods for solving the Toda lattice equations, emphasizing accuracy in capturing soliton peaks and oscillatory solutions, and highlights potential overestimations when benchmarking on idealized data.

Contribution

It provides a comprehensive benchmark of numerical methods for the Toda lattice, revealing limitations of current approaches and the importance of diverse initial data in accuracy assessment.

Findings

Benchmarking on pure-soliton data can overestimate accuracy.

Numerical inverse scattering transform provides high-accuracy reference solutions.

Different methods vary significantly in capturing oscillatory parts of solutions.

Abstract

We compare performances of well-known numerical time-stepping methods that are widely used to compute solutions of the doubly-infinite Fermi-Pasta-Ulam-Tsingou (FPUT) lattice equations. The methods are benchmarked according to (1) their accuracy in capturing the soliton peaks and (2) in capturing highly-oscillatory parts of the solutions of the Toda lattice resulting from a variety of initial data. The numerical inverse scattering transform method is used to compute a reference solution with high accuracy. We find that benchmarking a numerical method on pure-soliton initial data can lead one to overestimate the accuracy of the method.