Numerical simulation of language interactions using online coupled generalized multiscale finite element method. (English) Zbl 1505.65259
Summary: In this paper, we present a new mathematical model of the interaction of two languages. In the model, we distinguish percentages of people who speak a non-target language, a target language with low and high proficiency, and both languages considering the low and high proficiency levels in the target language. Therefore, the solution consists of five fields. Furthermore, we assume the diffusive and convective spread of the languages, considering the overflow between them. Thus, the mathematical model is defined by a coupled system of partial differential equations for the five fields. Since the mathematical model is coupled and the medium is heterogeneous, we have implemented a multiscale method. The proposed multiscale method is based on the Generalized Multiscale Finite Element Method (GMsFEM). In addition to offline multiscale basis functions, we also construct online multiscale basis functions. The online basis functions can account for changes in the heterogeneity of the medium caused by migration flows. Numerical results have shown that such online enrichment can significantly improve the accuracy of multiscale modeling.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 68T50 | Natural language processing |
| 91F20 | Linguistics |
Keywords:
language interaction; sociolinguistics; interdisciplinary research; mathematical modeling; generalized multiscale finite element method; online basis functionsReferences:
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