Minimal $\ell^2$ Norm Discrete Multiplier Method

TL;DR

This paper presents the Minimal $ ext{l}^2$ Norm Discrete Multiplier Method (MN-DMM), an extension of DMM that efficiently constructs conservative finite difference schemes for complex dynamical systems with multiple conserved quantities.

Contribution

The paper introduces MN-DMM, which uses the Moore-Penrose pseudoinverse to solve underdetermined problems, simplifying the construction of conservative schemes for large systems with multiple invariants.

Findings

MN-DMM is consistent and conservative.

Numerical examples demonstrate wide applicability.

Variants like SVD-based MN-DMM improve practical implementation.

Abstract

We introduce an extension to the Discrete Multiplier Method (DMM), called Minimal $\ell_2$ Norm Discrete Multiplier Method (MN-DMM), where conservative finite difference schemes for dynamical systems with multiple conserved quantities are constructed procedurally, instead of analytically as in the original DMM. For large dynamical systems with multiple conserved quantities, MN-DMM alleviates difficulties that can arise with the original DMM at constructing conservative schemes which satisfies the discrete multiplier conditions. In particular, MN-DMM utilizes the right Moore-Penrose pseudoinverse of the discrete multiplier matrix to solve an underdetermined least-square problem associated with the discrete multiplier conditions. We prove consistency and conservative properties of the MN-DMM schemes. We also introduce two variants - Mixed MN-DMM and MN-DMM using Singular Value Decomposition - and discuss their usage in practice. Moreover, numerical examples on various problems arising from the mathematical sciences are shown to demonstrate the wide applicability of MN-DMM and its relative ease of implementation compared to the original DMM.