Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model

TL;DR

This paper develops implicit-explicit time discretization schemes for optimal control problems governed by the Goldstein-Taylor model, ensuring stability and accuracy in the diffusive limit, with numerical validation.

Contribution

It introduces asymptotic preserving IMEX schemes that remain stable and accurate for the heat equation limit in optimal control problems.

Findings

IMEX schemes provide stable discretization in the diffusive limit

Discrete optimality system aligns with the heat equation behavior

Numerical examples confirm theoretical predictions

Abstract

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior.