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Modal reduction of PDE models by means of snapshot archetypes. (English) Zbl 1040.35089

Summary: A new method for constructing low-dimensional reduced models of dissipative partial differential equations is proposed. The original PDE, \(u_t=F(u)\), is projected onto a linear subspace spanned by the so-called snapshot archetypes, that are selected spatial profiles of \(u(x,t)\). The selection rule of the snapshot archetypes characterizes the method. Two different selection methods are proposed. We provide an “energetic” criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics. This approach is tested for several PDE systems such as the Kuramoto-Sivashinsky equation, the Arneodo-Elezgaray reaction-diffusion model, and the self-ignition dynamics of a coal stockpile. The latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35J20 Variational methods for second-order elliptic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs

Software:

AUTO-86; AUTO
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