Recursive POD expansion for the advection-diffusion-reaction equation. (English) Zbl 1475.74125
Summary: This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. Azaïez et al., Commun. Comput. Phys. 24, No. 5, 1556–1578 (2018; Zbl 07416706)] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.
MSC:
| 74S22 | Isogeometric methods applied to problems in solid mechanics |
| 65D15 | Algorithms for approximation of functions |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
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