Many partial differential equations (PDEs) admit conserved quantities like mass or energy. Those quantities are often essential to establish well-posed results. When approximating a PDE by a finite-difference scheme, it is natural to ask whether related discretized quantities remain conserved under the scheme. Such conservation may establish the stability of the numerical scheme. We present an algorithm for checking the preservation of a polynomial quantity under a polynomial finite-difference scheme. In our algorithm, schemes can be explicit or implicit, have higher-order time and space derivatives, and an arbitrary number of variables. Additionally, we present an algorithm for, given a scheme, finding conserved quantities. We illustrate our algorithm by studying several finite-difference schemes.

In partial differential equations (PDEs), conserved quantities like mass and momentum are fundamental to understanding the behavior of the described physical systems. The preservation of conserved quantities is essential when using numerical schemes to approximate solutions of corresponding PDEs. If the discrete solutions obtained through these schemes fail to preserve the conserved quantities, they may be physically meaningless and unreliable. Previous approaches focused on checking conservation in PDEs and numerical schemes, but they did not give adequate attention to systematically handling parameters. This is a crucial aspect because many PDEs and numerical schemes have parameters that need to be dealt with systematically. Here, we investigate if the discrete analog of a conserved quantity is preserved under the solution induced by a parametric finite difference method. In this thesis, we modify and enhance a pre-existing algorithm to effectively and reliably deduce conserved quantities in the context of parametric schemes, using the concept of comprehensive Groebner systems. The main contribution of this work is the development of a versatile algorithm capable of handling various parametric explicit and implicit schemes, higher-order derivatives, and multiple spatial dimensions. The algorithm’s effectiveness and efficiency are demonstrated through examples and applications. In particular, we illustrate the process of selecting an appropriate numerical scheme among a family of potential discretization for a given PDE.