Partitioned Linear Multistep Methods (PLMM)

• Partitioned Linear Multistep Methods (PLMM) are advanced time integration schemes that decouple system components using separate multistep formulas for enhanced flexibility and invariant preservation.
• They leverage symmetry and time-reversibility to suppress oscillatory errors and maintain bounded global error growth in long-term simulations of Hamiltonian and dispersive systems.
• PLMM are practically applied to semidiscrete PDEs, such as the nonlinear Schrödinger and Boussinesq equations, where they effectively conserve invariants like mass, momentum, and energy.

Partitioned Linear Multistep Methods (PLMMs) are a class of time integration techniques for ordinary differential equations (ODEs) and semidiscrete partial differential equations (PDEs) in which the system is separated into distinct components, each advanced with potentially different multistep formulas. PLMMs offer flexibility in the treatment of multi-physics, non-separable Hamiltonian systems, and dispersive wave models, particularly when preservation of invariants and long-term qualitative accuracy are required. Their theoretical underpinnings, error properties, and structure-preserving advantages have been rigorously established for both ODE and semidiscrete PDE applications, especially in the context of symmetric (time-reversible) schemes.

1. Definition and Core Structure

Consider an autonomous ODE system in partitioned form: $$\dot P(t) = f(P(t), Q(t)), \quad \dot Q(t) = g(P(t), Q(t)), \quad P(t)\in\mathbb{R}^d,\; Q(t)\in\mathbb{R}^{n-d}.$$ A PLMM advances the discrete solution via two (generally irreducible) pairs of generating polynomials: $(\rho_p, \sigma_p)$ of degree $k_p$ for the $P$-component and $(\rho_q, \sigma_q)$ of degree $k_q$ for the $Q$-component: $$\rho_p(E) P_n = \Delta t\, \sigma_p(E) f(P_n, Q_n), \quad \rho_q(E) Q_n = \Delta t\, \sigma_q(E) g(P_n, Q_n),$$ with $E$ the shift operator: $(Ey)_n = y_{n+1}$. The polynomials take the form

$(\rho_p, \sigma_p)$0

Zero-stability and consistent order $(\rho_p, \sigma_p)$1 for both partitions are fundamental prerequisites. A start-up procedure provides sufficiently accurate initial values: $(\rho_p, \sigma_p)$2 PLMMs generalize standard linear multistep schemes by decoupling the update rules for distinct subsystems; this is particularly relevant for partitioned Hamiltonian systems and splitting applications.

2. Symmetry and Time-Reversibility

A single linear multistep method (LMM)

$(\rho_p, \sigma_p)$3

is termed symmetric if

$(\rho_p, \sigma_p)$4

implying that the local-error coefficients $(\rho_p, \sigma_p)$5 vanish for all odd $(\rho_p, \sigma_p)$6. A PLMM is symmetric if both pairs $(\rho_p, \sigma_p)$7 and $(\rho_p, \sigma_p)$8 satisfy the symmetry condition and the characteristic polynomials share no unit-modulus roots other than the simple root at $(\rho_p, \sigma_p)$9. Symmetry ensures time-reversibility and is critical for preservation of invariants and suppression of secular error drift.

The palindromic property of coefficients in symmetric methods guarantees that local truncation errors are exclusively of even order, leading to enhanced long-term error control for conserved quantities in Hamiltonian and reversible systems.

3. Invariant Preservation in Semidiscrete PDEs

PLMMs are especially adept at preserving discrete analogues of PDE invariants when applied after a spatial discretization. Two canonical case studies exemplify this:

(a) Nonlinear Schrödinger (NLS) Equation — Pseudospectral Discretization:

$k_p$0

where $k_p$1 is the spectral second-derivative. Invariants (mass $k_p$2, momentum $k_p$3, energy $k_p$4) are retained up to error terms quantifiable in terms of the PLMM order and symmetry properties.

(b) Boussinesq System:

$k_p$5

with invariants $k_p$6 explicitly constructed. When parameters satisfy $k_p$7, the system is Hamiltonian (in non-canonical form), and the same symmetry-driven invariant control applies.

In both settings, discrete invariants are algebraically expressed in terms of the system's variables and differentiation matrices resulting from pseudospectral collocation or finite difference approaches.

4. Long-Term Error and Conservation: Theoretical Results

For symmetric PLMMs, rigorous long-term error bounds have been established. The main theorems assert:

NLS Semidiscrete Case: If initial values are accurate to $k_p$8 (or $k_p$9 with improved start-up), then, for solitary-wave solutions,

$P$0

where $P$1 denotes any invariant mass, momentum, or energy.

Boussinesq Case: Under similar hypotheses, errors in $P$2 and $P$3 satisfy

$P$4

with sufficiently accurate start-up data.

These results fundamentally require symmetry, zero-stability, and no spurious unit-modulus roots (other than 1) in the generating polynomials, as well as sufficiently accurate spatial discretization.

5. Error Expansion and Oscillatory Mode Control

The analysis is anchored in a two-scale asymptotic expansion of the global error: $P$5 with analogous expansions for $P$6. Here, $P$7 are the (nontrivial) unit-modulus roots of $P$8. The "smooth" part $P$9 may grow polynomially with time, while oscillatory terms $(\rho_q, \sigma_q)$0 remain bounded if the associated linearized transition matrices are uniformly bounded.

For invariants $(\rho_q, \sigma_q)$1, error expansion yields, under symmetry and Hamiltonian structure, that leading error integrals involving higher derivatives can be integrated by parts to yield only boundary terms and oscillatory integrals that remain bounded for solitary wave solutions. The method thus effectively suppresses secular drift in invariants on long time intervals.

A sufficient criterion for the boundedness of parasitic (oscillatory) components is the absence of multiple unit-circle roots other than $(\rho_q, \sigma_q)$2; this avoids exponential amplification and ensures bounded error propagation.

6. Practical Implementation and Applications

PLMMs are widely used as time integrators in the semi-discrete setting arising from Fourier pseudospectral or high-order finite difference discretization of nonlinear dispersive PDEs over periodic domains. For instance, in the NLS case, spatial domain truncation to a long interval makes the solitary wave negligible at boundaries; the pseudospectral method yields high-accuracy spatial derivatives and a system for $(\rho_q, \sigma_q)$3 amenable to PLMM integration.

In Boussinesq-type systems with Hamiltonian structure (for $(\rho_q, \sigma_q)$4), PLMMs enable uniform-in-time bounds on discrete invariant errors, utilizing the explicit exponential decay of the solitary-wave profile.

Beyond dispersive wave problems, PLMMs have been adapted to stiff PDEs in applied mathematics, such as in splitting schemes for the Heston model in financial mathematics (Hundsdorfer et al., 2017), using stabilizing corrections and careful extrapolation for explicit and implicit partitions. In these cases, PLMMs (of BDF, Adams, and CNLF type) are competitive with classical ADI schemes regarding unconditional stability and error, provided appropriate parameter choices are made.

7. Numerical Evidence and Qualitative Insights

Empirical studies reinforce the theoretical predictions:

• For the NLS cubic soliton on $(\rho_q, \sigma_q)$5 with $(\rho_q, \sigma_q)$6, the symmetric 2-step PLMM (SPLMM2) ensures that invariant errors in mass and energy remain $(\rho_q, \sigma_q)$7 and bounded up to $(\rho_q, \sigma_q)$8, whereas non-symmetric methods show linear or exponential drift.
• Non-partitioned Adams3 methods exhibit secular (linear) growth of invariant error, in accordance with their lack of symmetry.
• In Boussinesq tests (Bona–Smith parameters), SPLMM2 produces "super-convergence" ($(\rho_q, \sigma_q)$9 error in $k_q$0), outperforming its nominal second-order status—an effect predicted by the error expansions.
• For perturbed solitons or evolving wave trains, SPLMM2 keeps all monitored errors (mass, momentum, energy) uniformly small and bounded over long times, underscoring the necessity of symmetry and invariant control for qualitative accuracy in nonlinear wave simulation.

A summary of leading energy error growth for various methods is as follows:

Method	Symmetry	Partitioned	Error Growth
PLMM₂	Yes	Yes	Bounded (oscill.)
LMM₂	Yes	No	Exponential
Adams₃	No	No	Linear
Mixed	No (q)	Yes	Linear after trans.

This demonstrates the critical role of symmetry and partitioning for guaranteed control of error growth and invariants.

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PLMMs form a versatile and theoretically robust framework for time integration in partitioned and stiff ODEs or semidiscrete PDEs, enabling the practitioner to combine explicitness, high order, and structure-preserving properties—features essential for the long-term simulation of Hamiltonian and dispersive systems (Cano et al., 4 Nov 2025, Cano et al., 2024, Hundsdorfer et al., 2017).