Benchmarking Numerical Methods for Lattice Equations with the Toda Lattice

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.