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Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction. (English) Zbl 1060.65105
The authors are interested in reaction-diffusion problems of the type \((U^\varepsilon \in\mathbb{R}^n)\): \[ \partial_tU^\varepsilon_x \cdot\bigl(B (U^\varepsilon) \partial_xU^\varepsilon\bigr)=F^\varepsilon,\quad x\in \mathbb{R}^d,\;t\geq 0 \] where \(B(U^\varepsilon)\), called diffusion matrix, is a tensor of order \(d\times d\times n\). The solution of this dynamical system is denoted by \(U^\varepsilon=T^t_\varepsilon U_0\), where \(U_0\) is the initial condition and \(\varepsilon>0\) is a small parameter. Then the singular perturbation results are presented followed by various operator splittings. The theoretical results are illustrated numerically together with the influence of the discretization on the results.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
Software:
ODEPACK
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