A new Lagrange multiplier approach for constructing structure preserving schemes, I. positivity preserving

TL;DR

This paper introduces a novel Lagrange multiplier method for creating positivity-preserving schemes for parabolic equations, combining predictor-corrector steps with stability analysis and broad applicability.

Contribution

It develops a general, efficient Lagrange multiplier framework for positivity and mass conservation in numerical schemes, applicable to various discretizations.

Findings

The proposed schemes are stable under general conditions.

Numerical tests confirm the effectiveness of the positivity preservation.

The approach is flexible and can be integrated with different time discretizations.

Abstract

We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.