Systematisation of Systems Solving Physics Boundary Value Problems

Abstract

A general conservation law that defines a class of physics field theories is constructed. First, the notion of a general field is introduced as a formal sum differential forms on a Minkowski manifold. By the action principle the conservation law is defined for such a general field. By construction, particular field notions of physics, e.g., magnetic flux, electric field strength, stress, strain etc. become instances of the general field. Hence, the differential equations that constitute physics field theories become also instances of the general conservation law. The general field and the general conservation law together correspond to a large class of relativistic hyperbolic physics field models. The parabolic and elliptic models can thereafter be derived by adding constraints. The approach creates solid foundations for developing software systems for scientific computing; the unifying structure shared by the class of field models makes it possible to implement software systems which are not restricted to certain predefined problems. The versatility of the proposed approach is demonstrated by numerical experiments with moving and deforming domains.

Time-harmonic electromagnetics with exact controllability and discrete exterior calculus

“…Further applications in quantum mechanics were then considered in [26][27][28]. The systematization of scientific software development covering fundamental conservation laws was sketched in [29], and a generic category-theoretic framework of spacetime linear wave phenomena was finally proposed by [30] in 2022. Such formalisms allow, e.g., moving and deforming computational domains [29].…”

“…The systematization of scientific software development covering fundamental conservation laws was sketched in [29], and a generic category-theoretic framework of spacetime linear wave phenomena was finally proposed by [30] in 2022. Such formalisms allow, e.g., moving and deforming computational domains [29]. Another research track considers higherorder discrete exterior calculus involving higher order Whitney forms and novel Hodge star formulations [31][32][33].…”