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Solving optimization-constrained differential equations with discontinuity points

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C. Landry, A. Caboussat and E. Hairer

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Abstract. Ordinary differential equations are coupled with constrained optimization problems when modeling a system at equilibrium evolving with time. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method for the resolution of optimization-constrained differential equations is proposed by coupling an implicit Runge-Kutta method (RADAU5), with numerical techniques for the detection of the events (activation and deactivation of constraints) when the system evolves with time. The computation of the events is based on dense output formulas, continuation techniques and geometric arguments. Numerical results are presented for the simulation of the time-dependent equilibrium of organic atmospheric aerosol particles, and show the efficiency and accuracy of the approach.

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Key Words. Initial value problems, differential-algebraic equations, constrained optimization, Runge-Kutta methods, event detection, discontinuity points, computational chemistry.