On the construction of partial difference schemes II: discrete variables and Schwarzian lattices
For researchers in numerical analysis and geometric integration, this work addresses the open problem of lattice non-commutativity in invariant difference schemes, but the results are incremental.
The paper presents a procedure for discretizing partial differential equations on arbitrary lattices and applies it to the potential Burgers equation, comparing an orthogonal lattice (satisfying commutativity) with an exponential lattice (violating it). Numerical results show different behaviors for two exact solutions.
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.