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Algorithmic detection of conserved quantities of finite-difference schemes for partial differential equationsKrannich, Friedemann 04 1900 (has links)
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.
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Deducting Conserved Quantities for Numerical Schemes using Parametric Groebner SystemsMajrashi, Bashayer 05 1900 (has links)
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.
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