On the construction of partial difference schemes II: discrete variables and Schwarzian lattices

TL;DR

This paper develops a procedure for discretizing partial differential equations on arbitrary lattices, analyzing the implications of non-invariant lattices that violate classical theorems, with numerical comparisons of different schemes.

Contribution

It introduces a method for constructing invariant difference schemes on arbitrary lattices, including non-invariant ones that do not satisfy the Clairaut--Schwarz--Young theorem.

Findings

Orthogonal lattice scheme preserves symmetries and shows stable numerical behavior.

Exponential lattice scheme, not invariant, exhibits different numerical properties.

Numerical results highlight the impact of lattice choice on solution accuracy.

Abstract

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut--Schwarz--Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut--Schwarz--Young theorem. A discussion on the numerical results is also presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.