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In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use a specific language (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous medium to test these approaches. The results show that both approaches can significantly reduce the discrete problem's size. However, the online approach achieves better accuracy. • We consider a language competition process between two languages in a heterogeneous medium. • The mathematical model considers diffusive-convective spatial spread, bilingualism, logistic growth, and diglossia. • For efficient modeling, we present offline and online multiscale approaches based on the GMsFEM and the Residual-Driven Online GMsFEM, respectively. • Numerical results show that both multiscale approaches have high accuracy, but the online approach provides better results than the offline one. • The online adaptive strategy provides accuracy similar to the uniform one but with fewer degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new mathematical model of language interactions considering bilingualism. We assume diffusive and convective language spreads with language exchange terms. The resulting model consists of a coupled system of partial differential equations for three functions. We develop a finite element approximation of the mathematical model using implicit and partially explicit time schemes. In addition, we present a partial learning approach for this problem using a partially explicit discretization. In this method, we train a Deep Neural Network to predict the values of the difficult-to-compute (implicit) part of the solution at some observed points. We then perform linear temporal interpolation and spatial interpolation using the Proper Orthogonal Decomposition (POD) and the Discrete Empirical Interpolation Method (DEIM). To test the proposed approach and the model, we consider two model problems. Each of them simulates different language situations. For each problem, we compute the relative errors of the proposed approach and define the different parts of the error. The numerical results show that the proposed approach can provide good accuracy while reducing computational costs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mathematical model of the interaction of two languages. In our model, we consider two languages, though it can be generalized to multiple languages, which compete in a heterogeneous environment, consisting of highly varying properties related to the dynamics of the interaction. We use coupled convection–diffusion–reaction equations to describe the processes. Each equation describes the dynamics of one of the languages and contains terms related to the stand-alone dynamics and some coupling terms. The coupling terms represent the interaction between languages. We propose a numerical approach for solving the proposed model equation. In particular, we consider various inhomogeneities associated with cities and countryside, where languages are used differently (e.g., Sakha republic). These dynamics are essential for understanding the evolution of languages (one being dominant) and linguistic ecology that studies languages and their use in real/social life. Because of heterogeneities associated with geography, we use a multiscale approach. The proposed multiscale approach designs special basis functions to represent the small-scale information on larger scales. This way, we can solve the problem on a much coarser grid. Numerical results are presented that describe the dynamics and interaction of two languages. The main novelty of the paper consists of the proposed model and a multiscale algorithm. [ABSTRACT FROM AUTHOR]