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A Structure-Preserving Numerical Scheme for Optimal Control and Design of Mixing in Incompressible Flows

Weiwei Hu, Ziqian Li, Yubiao Zhang, Enrique Zuazua

TL;DR

This work tackles optimal mixing of a passive scalar in incompressible flows by developing a structure-preserving numerical framework that exactly conserves mass and $L^2$-energy and preserves state--adjoint duality at the discrete level. The method combines a centered finite-volume spatial discretization with a time-symmetric Crank--Nicolson scheme for both the forward transport and its adjoint, integrated into a gradient-based optimization over a finite set of stirring modes. The authors prove that the discrete scheme preserves key invariants and duality and that Optimize-Then-Discretize and Discretize-Then-Optimize formulations are equivalent, ensuring reliable gradient information. Numerical experiments show that time-dependent optimization across a small basis of stirring modes yields near-exponential decay of the mix-norm, substantially outperforming any single steady mode while maintaining exact conservation, suggesting a practical design principle for actuator-constrained mixing in complex geometries.

Abstract

We develop a structure-preserving computational framework for optimal mixing control in incompressible flows. Our approach exactly conserves the continuous system's key invariants (mass and $L^2$-energy), while also maintaining discrete state-adjoint duality at every time step. These properties are achieved by integrating a centered finite-volume discretization in space with a time-symmetric Crank-Nicolson integrator for both the forward advection and its adjoint, all inside a gradient-based optimization loop. The result is a numerical solver that is faithful to the continuous optimality conditions and efficiently computes mixing-enhancing controls. In our numerical tests, the optimized time-dependent stirring produces a nearly exponential decay of a chosen mix-norm, achieving orders-of-magnitude faster mixing than any single steady flow. To our knowledge, this work provides the first evidence that enforcing physical structure at the discrete level can lead to both exact conservation and highly effective mixing outcomes in optimal flow design.

A Structure-Preserving Numerical Scheme for Optimal Control and Design of Mixing in Incompressible Flows

TL;DR

This work tackles optimal mixing of a passive scalar in incompressible flows by developing a structure-preserving numerical framework that exactly conserves mass and -energy and preserves state--adjoint duality at the discrete level. The method combines a centered finite-volume spatial discretization with a time-symmetric Crank--Nicolson scheme for both the forward transport and its adjoint, integrated into a gradient-based optimization over a finite set of stirring modes. The authors prove that the discrete scheme preserves key invariants and duality and that Optimize-Then-Discretize and Discretize-Then-Optimize formulations are equivalent, ensuring reliable gradient information. Numerical experiments show that time-dependent optimization across a small basis of stirring modes yields near-exponential decay of the mix-norm, substantially outperforming any single steady mode while maintaining exact conservation, suggesting a practical design principle for actuator-constrained mixing in complex geometries.

Abstract

We develop a structure-preserving computational framework for optimal mixing control in incompressible flows. Our approach exactly conserves the continuous system's key invariants (mass and -energy), while also maintaining discrete state-adjoint duality at every time step. These properties are achieved by integrating a centered finite-volume discretization in space with a time-symmetric Crank-Nicolson integrator for both the forward advection and its adjoint, all inside a gradient-based optimization loop. The result is a numerical solver that is faithful to the continuous optimality conditions and efficiently computes mixing-enhancing controls. In our numerical tests, the optimized time-dependent stirring produces a nearly exponential decay of a chosen mix-norm, achieving orders-of-magnitude faster mixing than any single steady flow. To our knowledge, this work provides the first evidence that enforcing physical structure at the discrete level can lead to both exact conservation and highly effective mixing outcomes in optimal flow design.
Paper Structure (32 sections, 7 theorems, 129 equations, 20 figures, 1 algorithm)

This paper contains 32 sections, 7 theorems, 129 equations, 20 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Computational challenges and motivations.
  4. Main results and contributions
  5. Related work
  6. Organization of the paper
  7. Numerical scheme
  8. Full discretization of the state equation
  9. Computation of the Lagrange multiplier
  10. Backward discretization of the adjoint equation.
  11. Numerical scheme for problem (OP)
  12. Structure-preserving properties of the numerical scheme
  13. Commutativity of Optimization and Discretization
  14. Numerical experiments
  15. Cellular flow basis
  16. ...and 17 more sections

Key Result

Theorem 3.1

Assume that the fluxes satisfy the exact conditions in Remark rem:exact_flux_conditions. Let $\theta^{(n)}_h$ and $\rho^{(n)}_h$ be the solutions at time level $t_n$ ($n \in \{0,1,\ldots, N_t\}$) to equations eq:CN_cell and 20251004-DiscreteAdjointEquation with arbitrary initial data $\theta_h^{(0)}

Figures (20)

  • Figure 5.1: Cellular basis flows $\mathbf{b}_1$ and $\mathbf{b}_2$ defined in \ref{['eq:cellular']}.
  • Figure 5.2: Evolution of $\Vert \theta_h \Vert_{\dot{H}^{-1}(\Omega)}$ with cellular flow $\mathbf{b}_1$ under initial data \ref{['eq:theta_init_1']} and \ref{['eq:theta_init_2']}.
  • Figure 5.3: Evolution of $\theta_h$ with cellular flow $\mathbf{b}_1$ and initial data \ref{['eq:theta_init_1']}.
  • Figure 5.4: Evolution of $\theta_h$ with cellular flow $\mathbf{b}_1$ and initial data \ref{['eq:theta_init_2']}.
  • Figure 5.5: Evolutions of $v_1(t)$, $v_2(t)$, mix-norm $\Vert \theta_h \Vert_{\dot{H}^{-1}(\Omega)}$, mass $M_h(\theta_h)$, energy $E_h(\theta_h)$ and state-adjoint pairing $\langle \theta_h, \rho_h \rangle_{X_h}$ for $t\in[0, 1]$ with initial control \ref{['eq:u1u2_1']} and initial data \ref{['eq:theta_init_1']}.
  • ...and 15 more figures

Theorems & Definitions (18)

  • Remark 2.1: Exact flux conditions for structure preservation
  • Remark 2.2: Why not an upwind form
  • Remark 2.3: Spurious oscillations
  • Remark 2.4: Algebraic adjoint interpretation
  • Theorem 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['20251008-theorem-PerservedStructures']}
  • Theorem 4.1
  • proof
  • Lemma 7.1
  • ...and 8 more