Invariant Discretization of Partial Differential Equations Admitting Infinite-Dimensional Symmetry Groups
Provides a method for preserving symmetries in numerical discretizations of PDEs, relevant to researchers in numerical analysis and mathematical physics.
The paper addresses the challenge of constructing invariant numerical schemes for PDEs with infinite-dimensional symmetry groups, proposing discretization of the symmetry pseudo-group action using moving frames. Computer simulations show that the resulting schemes can outperform standard ones.
Given a differential equation with infinite-dimensional symmetry pseudo-group it is shown, using an example, that it is generally not possible to construct enough joint invariants to form an invariant numerical scheme of the equation. To circumvent this problem, we propose to discretize the symmetry pseudo-group action. Using the theory of moving frames, joint invariants of the discretized action are algorithmically constructed. Computer simulations indicate that numerical schemes constructed from these joint invariants can produce better numerical results than standard schemes.