Numerical solution of hyperbolic equations using the finite element method for modeling dynamic systems

In this study, the state of the art in solving hyperbolic equations using the finite element method has been analyzed in order to achieve accurate and stable numerical solutions. Various shape functions and variational formulations have been investigated to assess their influence on the accuracy and stability of computational algorithms. The advantage of hybrid finite-element schemes in modeling wave processes in heterogeneous media has been identified. The effect of time-step selection and mesh adaptation on solution convergence and computational cost has been examined. An optimal relationship between the order of approximation and the degree of mesh refinement has been determined to improve accuracy under limited computational resources. It has been established that the proposed algorithm ensures a reliable numerical representation of a wide range of wave dynamics and continuum mechanics problems. A justification for the applicability of optimization methods for finite element method parameters in the solution of multidimensional hyperbolic problems has been provided for further integration into industrial and research software packages.

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