Quasi-Orthogonal Runge-Kutta Projection Methods

TL;DR

This paper introduces a quasi-orthogonal projection method for explicit Runge-Kutta schemes that preserves invariants and accuracy with minimal computational overhead, improving stability and physical fidelity in numerical simulations.

Contribution

The paper presents a novel quasi-orthogonal projection technique for explicit RK methods that maintains invariants and accuracy efficiently, applicable to various systems with invariants.

Findings

Preserves linear invariants during simulations.

Maintains the order of accuracy of the base RK method.

Demonstrates effectiveness across multiple applications.

Abstract

A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under exact solution of their governing PDEs. However, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce these constraints, leading to a loss of accuracy and stability. Projection is an efficient way to address this shortcoming by correcting the RK solution at the end of each time step. Here we introduce a novel projection method for explicit RK schemes, called a \textit{quasi-orthogonal} projection method. This method can be employed for systems containing a single (not necessarily convex) invariant functional, for dissipative systems, and for the systems containing multiple invariants. It works by projecting the orthogonal search direction(s) into the solution space spanned by the RK stage derivatives. With this approach linear invariants of the problem are preserved, the time step size remains fixed, additional computational cost is minimal, and these optimal search direction(s) preserve the order of accuracy of the base RK method. This presents significant advantages over existing projection methods. Numerical results demonstrate that these properties are observed in practice for a range of applications.