Algorithmic Detection of Conserved Quantities for Finite-Difference Schemes

Many partial differential equations (PDEs) admit conserved quantities such as mass for the heat equation or energy for the advection equation. These are often essential for establishing well-posedness results. When approximating a PDE with a finite-difference scheme, it is crucial to determine whether related discretized quantities remain conserved by the scheme. Such conservation may ensure the stability of the numerical scheme. We present an algorithm for verifying the preservation of a polynomial quantity under a polynomial finite-difference scheme. Our schemes can be explicit or implicit, have higher-order time and space derivatives, and have an arbitrary number of variables. Additionally, we introduce an algorithm for finding conserved quantities. We illustrate our algorithm in several finite-difference schemes. Our approach incorporates a naive implementation of Comprehensive Gröbner Systems to handle parameters, ensuring accurate computation of conserved quantities.