Local Discontinuous Galerkin method for the heterodimer model of protein interaction
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In the recent work [2], a Local Discontinuous Galerkin method based the “boundedness-by-entropy” framework in [1] for nonlinear cross-diffusion systems was introduced. Such a method is endowed with several valuable properties, such as i) allowing for an arbitrary degree of approximation in space, ii) preserving the positivity or boundedness of the physical model, iii) permitting an efficient computation of the nonlinear terms, and iv) satisfying a discrete version of the entropy-stability estimate of the continuous model. In this talk, we discuss the main theoretical and computational aspects of the LDG method in [2] applied to the heterodimer model, which consists of two coupled semilinear parabolic PDEs describing the evolution and mutual interaction of biological species. We pay special attention to the applications of the heterodimer model related to the progression of neurodegenerative diseases [3]. At the end of this talk, we present some physically-motivated simulations and show the effectiveness of the proposed approach.