Quasi-Orthogonal Runge-Kutta Projection Methods

TL;DR

The paper tackles preserving nonlinear invariants in time integration of semi-discrete PDEs by introducing a quasi-orthogonal projection for explicit Runge-Kutta schemes. The method constructs a projection direction from the RK stage subspace $\mathcal{S}=\mathrm{span}(\underline{R}_1,\dots,\underline{R}_s)$ via an orthogonal decomposition of $\nabla G(\underline{q}^{n+1})$ and updates $\hat{\underline{q}}^{n+1}=\underline{q}^{n+1}+\lambda_n\frac{\nabla G_s}{\|\nabla G_s\|_2}$ to enforce the invariant $G(\hat{\underline{q}}^{n+1})=G(\underline{q}^n)$. This approach preserves linear invariants and maintains the base RK order without step-size relaxation, while extending to dissipative systems (monotonicity) and multiple invariants (via a nonsingular matrix $M$). Numerical tests on linear dissipative systems, nonlinear oscillators, Burgers equation, and rigid-body rotation confirm invariant preservation and at least the original order of accuracy, with minimal extra cost. The results indicate a robust, efficient path to structure-preserving time integration for explicit RK methods in a broad class of problems, with future work aimed at implicit and IMEX extensions.

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.

Figures (12)

• Figure 1: Visualization of how each of the projection techniques project RK solution to the nonlinear invariant manifold
• Figure 2: Energy change after the first time step for energy dissipative ODE system (\ref{['Eq_lin_decay']}) integrated with standard RK(4,4), relaxation method with RK(4,4), and quasi-orthogonal projection method with RK(4,4).
• Figure 3: Actual time step size over input step size for relaxation and quasi-orthogonal projection methods after the first time step integrating ODE system (\ref{['Eq_lin_decay']}).
• Figure 4: Energy evolution in the nonlinear oscillator problem (\ref{['Eq_nonlin_oscil']}) integrated with different integration schemes and a time step of $\Delta t=0.1$. All the original RK methods change energy up to their truncation error, but with the relaxation and quasi-orthogonal projection methods energy is preserved up to machine precision.
• Figure 5: Convergence study for the nonlinear oscillator problem (\ref{['Eq_nonlin_oscil']}). For base RK methods convergence is demonstrated by solid lines, while for their quasi-orthogonal projection counterpart it is depicted by dashed lines. With the proposed method, order of accuracy is equal to, or higher than, the corresponding base RK scheme.

Theorems & Definitions (25)

• Remark 1
• Lemma 1
• proof : Proof of Lemma \ref{['nabla_G_biggest_innerprod']}
• Lemma 2
• proof : Proof of Lemma \ref{['d_for_convex_G']}
• Corollary 2.1
• proof : Proof of Corollary \ref{['accuracy_directional_convex']}
• Remark 2
• Theorem 3
• proof : Proof of Theorem \ref{['Theorem_convexG']}