A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems

Rong Huang; Zhifeng Weng; Rong Huang; Zhifeng Weng

doi:10.3934/nhm.2023024

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 562-580. doi: 10.3934/nhm.2023024

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A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems

Rong Huang ,
Zhifeng Weng ^,

Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received: 05 November 2022 Revised: 14 January 2023 Accepted: 16 January 2023 Published: 31 January 2023

This paper describes a study of the barycentric interpolation collocation method for the optimal control problem governed by a nonlinear convection-diffusion equation. Using Lagrangian multipliers, we obtain the continuous optimality system which is composed of state equations, adjoint equations and optimality conditions. Then, barycentric interpolation collocation methods are applied to discretize the optimality system and the nonlinear term is treated by Newton's iteration. Furthermore, the corresponding consistency analyses of discrete schemes are demonstrated. Finally, the validity of the proposed schemes is demonstrated through several numerical experiments. Compared with the classical finite difference method, collocation schemes can yield the higher-order accurate solutions with fewer nodes.
- convection-diffusion equation,
- optimal control,
- barycentric interpolation collocation,
- finite difference
Citation: Rong Huang, Zhifeng Weng. A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems[J]. Networks and Heterogeneous Media, 2023, 18(2): 562-580. doi: 10.3934/nhm.2023024

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Abstract

This paper describes a study of the barycentric interpolation collocation method for the optimal control problem governed by a nonlinear convection-diffusion equation. Using Lagrangian multipliers, we obtain the continuous optimality system which is composed of state equations, adjoint equations and optimality conditions. Then, barycentric interpolation collocation methods are applied to discretize the optimality system and the nonlinear term is treated by Newton's iteration. Furthermore, the corresponding consistency analyses of discrete schemes are demonstrated. Finally, the validity of the proposed schemes is demonstrated through several numerical experiments. Compared with the classical finite difference method, collocation schemes can yield the higher-order accurate solutions with fewer nodes.

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