Convex Time-Stepping Integration

• Convex time-stepping is a numerical integration framework that formulates each time step as a convex minimization, ensuring unique, stable, and globally consistent updates.
• It incorporates embedded error estimators using step-doubling to adaptively adjust the time step, achieving O(δt²) local accuracy and reliable convergence.
• The method excels in simulating contact-rich, stiff systems by efficiently managing constraints and delivering real-time performance in complex hybrid dynamics.

Convex time-stepping refers to a numerical integration framework for dynamical systems in which each time step update is formulated as a convex optimization problem, with error control mechanisms that enforce prescribed accuracy on the discrete solution. This paradigm achieves global consistency and robustness, particularly for stiff and contact-rich systems, by leveraging convexity for solver reliability and incorporating embedded error estimators for adaptive step-size selection. Recent advances have demonstrated that convex time-stepping can match, and in many regimes outperform, classical error-controlled continuous integrators while providing rigorous accuracy guarantees and real-time performance for complex hybrid dynamical models (Kurtz et al., 11 Nov 2025).

1. Mathematical Formulation of Convex Time-Stepping

In the convex time-stepping approach, the state update over a time interval $\delta t$ is encoded as the solution to an unconstrained convex minimization problem that encapsulates all relevant physical phenomena—mass, inertia, contact, friction, holonomic constraints, actuation dynamics, and external signals. For multibody contact systems, the update for generalized velocities $v$ is

$$v^{n+1} = \arg\min_{v} \Big[ \tfrac{1}{2} v^T M v - r^T v + \sum_i \ell_{c,i}(v) + \sum_j \ell_{q,j}(v) + \ell_{\text{act}}(v) \Big]$$

where $M$ is the mass matrix evaluated at the current generalized coordinates, $r$ is the momentum plus discrete forces, $\ell_{c,i}(v)$ are compliant contact/friction potentials, $\ell_{q,j}(v)$ encode near-rigid constraints, and $\ell_\text{act}(v)$ models actuation (Kurtz et al., 11 Nov 2025). By construction, the minimization landscape is strictly convex, ensuring the global existence and uniqueness of each discrete update.

2. Error Estimation and Adaptive Step Control

Convex time-stepping incorporates an embedded error estimator for continuous accuracy enforcement. The canonical procedure in CENIC is step-doubling: compute two successive half-steps and one full step, then take the difference as the estimated local truncation error

$$e^{n+1} = \| S [q^{n+1} - \hat{q}^{n+1}] \|_{\infty}$$

Here, $S$ is a scaling matrix ensuring dimensionless error, $q^{n+1}$ is the position after two half-steps, and $\hat{q}^{n+1}$ is after a full step. The estimated error $e^{n+1}$ is used to adapt the next $\delta t$ by

$$\delta t_{\mathrm{new}} = \delta t \left(\frac{\epsilon_{\mathrm{acc}}}{e^{n+1}} \right)^{1/p}$$

with $p$ the order of the estimator (typically $p=1$ for step-doubling) (Kurtz et al., 11 Nov 2025). Step acceptance, rejection, and adaptation proceed via dead-band filtering and bounded growth policies, maintaining accuracy without excessive interval fluctuation.

3. Robustness, Convergence, and Theoretical Guarantees

Convex time-stepping delivers rigorous consistency and stability properties. The convexity of each update guarantees monotonic Newton solver convergence with no step failures, regardless of problem stiffness or contact complexity. The step-doubling error estimator quantifies $O(\delta t^2)$ local error, and accepted steps produce global solutions converging to the underlying DAE as $\delta t \to 0$.

For contact-rich, stiff systems—Hunt–Crossley compliant normal forces, regularized Coulomb friction, near-rigid holonomic constraints—convex potentials enforce robust singular perturbation regularization, damping transient constraint violations in $O(\beta \delta t)$ time (Kurtz et al., 11 Nov 2025). Moreover, symplectic IMEX schemes guarantee L-stability for the preferred first-order variant, ensuring rapid decay of high-frequency modes.

4. Algorithmic Details, Solver Integration, and Performance

Each time step comprises a convex quadratic minimization, typically solved via inexact Newton iterations with exact line search. Key performance optimizations:

• Warm-starting from previous iterates and between step-doubling/trapezoid variants.
• Adaptive convergence tolerances tied to requested $\epsilon_{\mathrm{acc}}$.
• Hessian factorization reuse, exploiting sparsity (mass matrix block-diagonal, separable contact potentials).
• Cubic line-search pre-initialization for rapid reduction (Kurtz et al., 11 Nov 2025).

Work-per-step is dominated by geometric queries and Newton solves (3–6 iterations per convex update, 3–9 solves per step). Empirically, CENIC achieves real-time rates ($>$100%) at moderate tolerances ($\epsilon_{\mathrm{acc}} \sim 10^{-3}$), outperforming error-controlled classical ODE integrators (RK3, SDIRK2, Rosenbrock) by $5$-$10\times$, as well as discrete-time engines (MuJoCo, Drake, Isaac Sim) except when the latter use very small fixed $\delta t$.

Method	$\epsilon=10^{-3}$ Wall-time	Real-Time Factor (%)
CENIC (sparse)	0.65 s	342
MuJoCo ($\Delta t=1$ ms)	0.45 s	21
RK3	7.50 s	—

5. Application Domains and Comparative Evaluation

Convex time-stepping excels in simulation of contact-heavy and hybrid dynamical systems typical in robotic manipulation, soft and hard multi-object clutter, and hybrid control. Its artifact-free integration is especially critical in scenarios with impacts, joint limit enforcement, and increasingly demanding actuator models. Quantitative work–precision profiles demonstrate superiority at loose and moderate tolerances for manipulation, grasping, object placement, and teleoperation replay tasks (Kurtz et al., 11 Nov 2025).

Artifact-free simulation is confirmed in dish-rack teleoperation demos: CENIC at $\epsilon=10^{-3}$ displays correct object behavior at high real-time rates, while discrete engines require $\Delta t \leq 1$ ms (with massive slowdowns) to suppress artifacts.

6. Limitations, Extensions, and Open Research Topics

While convex time-stepping achieves robust accuracy control and solver reliability, practical constraints include:

• First-order convergence in step-doubling, necessitating more steps at tight tolerances. Second-order trapezoid variants exist, but suffer lack of L-stability.
• Per-step convex optimization overhead (sparse linear algebra).
• Current implementations are single-threaded (future GPU/batch extensions needed).

Promising future directions involve L-stable second-order integrators with embedded error estimation, multi-rate splitting to treat contacts implicitly (reducing cost), and differentiable convex time-stepping for policy search and learning loops (Kurtz et al., 11 Nov 2025).

7. Synthesis: Convex Time-Stepping and Error-Controlled Integration

Convex time-stepping synthesizes the benefits of continuous error-controlled integration—guaranteed accuracy, adaptive stepping, robust stability—with the reliability and global convergence of convex optimization. The step-wise convex minimization ensures artifact-free handling of stiff contacts and constraints, while embedded error estimators maintain prescribed accuracy. This framework advances numerical integration of contact-rich mechanical systems, providing a bridge between high-fidelity physics and real-time simulation demands (Kurtz et al., 11 Nov 2025).