Optimal control using flux potentials: A way to construct bound-preserving finite element schemes for conservation laws

TL;DR

This paper introduces a novel flux potential-based optimization method for finite element schemes that preserves bounds in conservation laws without extra constraints, improving accuracy and efficiency.

Contribution

The paper proposes a new flux potential approach for bound-preserving finite element schemes, reducing unknowns and guaranteeing conservation without additional equality constraints.

Findings

Flux potentials reduce the number of unknowns in multidimensional problems.

The proposed method guarantees conservation and bounds preservation.

Numerical results show superiority over traditional flux limiting.

Abstract

To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal-dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear advection and anisotropic diffusion problems in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.