[Paper Review] Initializing and stabilizing variational multistep algorithms for modeling dynamical systems

C. Leland Ellison, J. W. Burby+3 more|arXiv (Cornell University)|Mar 4, 2014

Numerical methods for differential equations24 references3 citations

TL;DR

This paper proposes backward error initialization and periodic re-initialization to stabilize variational multistep integrators for dynamical systems with degenerate Lagrangians. By sampling initial conditions along solutions of the smooth modified system derived via backward error analysis, parasitic mode amplitudes are minimized, leading to significantly smoother conserved quantities and improved long-term numerical fidelity compared to standard one-step method initialization.

ABSTRACT

Backward error initialization and parasitic mode control are well-suited for use in algorithms that arise from a discrete variational principle on phase-space dynamics. Dynamical systems described by degenerate Lagrangians, such as those occurring in phase-space action principles, lead to variational multistep algorithms for the integration of first-order differential equations. As multistep algorithms, an initialization procedure must be chosen and the stability of parasitic modes assessed. The conventional selection of initial conditions using accurate one-step methods does not yield the best numerical performance for smoothness and stability. Instead, backward error initialization identifies a set of initial conditions that minimize the amplitude of undesirable parasitic modes. This issue is especially important in the context of structure-preserving multistep algorithms where numerical damping of the parasitic modes would violate the conservation properties. In the presence of growing parasitic modes, the algorithm may also be periodically re-initialized to prevent the undesired mode from reaching large amplitude. Numerical examples of variational multistep algorithms are presented in which the backward error initialized trajectories outperform those initialized using highly accurate approximations of the true solution.

Motivation & Objective

• Address the instability and poor smoothness of variational multistep algorithms due to parasitic modes arising from non-physical eigenvalues.
• Overcome the limitations of standard initialization using one-step methods, which fail to minimize parasitic mode amplitudes despite high accuracy.
• Develop a practical, structure-preserving initialization technique that maintains conservation properties while reducing numerical artifacts.
• Introduce periodic re-initialization using backward error analysis to control growing parasitic modes without violating conservation laws.
• Demonstrate the superiority of backward error initialization over conventional methods in long-term simulations of non-canonical Hamiltonian systems.

Proposed method

• Apply backward error analysis to derive the smooth modified system of a multistep variational integrator, retaining terms up to 𝒪(h²) for improved accuracy.
• Use the smooth modified system to generate initial conditions that minimize the amplitude of parasitic modes by aligning them with the desired physical trajectory.
• Implement periodic re-initialization by solving the smooth modified system with a one-step method to reset parasitic mode amplitudes to near-zero levels.
• Formulate the modified system as a perturbation of the original ODE: ẋ̃ = f(x̃) + h²[−1/24 f''(x̃)f(x̃)f(x̃) + 1/12 f'(x̃)f'(x̃)f(x̃)], capturing dominant parasitic mode behavior.
• Use symbolic or automatic differentiation to compute higher-order derivatives of f required for the modified system, enabling accurate initialization.
• Compare trajectories initialized via backward error analysis with those from high-order one-step methods (e.g., second-order Runge-Kutta), evaluating smoothness of conserved quantities.

Experimental results

Research questions

• RQ1 Can backward error initialization reduce parasitic mode amplitudes in variational multistep integrators more effectively than standard one-step method initialization?
• RQ2 How does the smooth modified system derived via backward error analysis improve the long-term behavior of multistep algorithms in non-canonical Hamiltonian systems?
• RQ3 To what extent does periodic re-initialization using the smooth modified system suppress growing parasitic modes without compromising conservation properties?
• RQ4 What is the quantitative impact of backward error initialization on the smoothness of conserved quantities like energy in systems such as planar point vortices?
• RQ5 How does the computational cost of backward error initialization compare to standard initialization, and is the performance gain justified?

Key findings

• Backward error initialization produces trajectories with undetectably small parasitic oscillations, while standard one-step initialization still exhibits visible oscillations even with high-accuracy approximations.
• The two-step midpoint method, when backward error initialized, shows significantly smoother conserved quantities compared to Runge-Kutta-initialized counterparts, despite both using similar initial accuracy.
• The parasitic mode amplitude is minimized when initial conditions lie along solutions of the smooth modified system, as derived from backward error analysis.
• Periodic re-initialization using the smooth modified system effectively resets parasitic modes to low amplitude, preventing their growth over long integration times.
• The modified system for the two-step midpoint method is explicitly derived as ẋ̃ = f(x̃) + h²[−1/24 f''(x̃)f(x̃)f(x̃) + 1/12 f'(x̃)f'(x̃)f(x̃)], which captures the dominant parasitic mode dynamics.
• Despite requiring higher-order derivatives and additional computation, backward error initialization is computationally viable due to its infrequent use relative to standard multistep updates.