---
title: Variable-Step DLN Time Integrator
url: https://www.emergentmind.com/topics/variable-step-dahlquist-liniger-nevanlinna-dln-time-integrator-3be2d178-6f64-4a77-9fd6-3877e2b9fc4c
type: topic
---

# Variable-Step DLN Time Integrator

The variable-step Dahlquist–Liniger–Nevanlinna (DLN) time integrator is a one-leg, two-step multistep method designed for unconditionally stable and second-order accurate time discretization of ordinary and partial differential equations with variable or adaptive time steps. The method is distinguished by its uniform $G$-stability, robust handling of non-uniform time grids, and ability to avoid nonphysical energy growth even under aggressive adaptivity. High-performance applications include incompressible flow, phase-field models, and nonlinear active fluid systems, especially when coupled with divergence-free mixed finite element spatial discretizations or minimum-dissipation adaptivity strategies [2510.16860][2509.19053][2507.21392][2409.19481][2306.02461][2407.19101][2007.03801][2001.08640][2108.09339].

## 1. General Formulation and Coefficient Structure

The DLN time integrator discretizes the ODE system $y'(t) = f(t, y(t))$ using an arbitrary, possibly adaptive partition $0 = t_0 < t_1 < \cdots < t_M = T$. Time steps are denoted $k_n = t_n - t_{n-1}$, with the step-ratio parameter
\[
\epsilon_n = \frac{k_n - k_{n-1}}{k_n + k_{n-1}},\quad r_n = \frac{k_n}{k_{n-1}},\quad \epsilon_n = \frac{r_n - 1}{r_n + 1}.
\]
The method uses the recurrence
\[
\alpha_0 y_{n-1} + \alpha_1 y_n + \alpha_2 y_{n+1} = \widehat k_n\, f(t_{n,\beta}, y_{n,\beta}),
\]
where the coefficients depend only on a user-prescribed parameter $\theta \in [0,1]$:
\[
\alpha_2 = \frac{1+\theta}{2},\quad \alpha_1 = -\theta,\quad \alpha_0 = \frac{\theta-1}{2},\qquad
\widehat k_n = \alpha_2 k_n - \alpha_0 k_{n-1}.
\]
The stage values $(t_{n,\beta}, y_{n,\beta})$ are convex combinations of $(t_{n-1}, t_n, t_{n+1})$ and $(y_{n-1}, y_n, y_{n+1})$ with weights $(\beta_0^{(n)}, \beta_1^{(n)}, \beta_2^{(n)})$ determined by $\theta$ and $\epsilon_n$. These guarantee second-order accuracy and $G$-stability for arbitrary step sequences. Uniform time steps recover classical schemes: $\theta=1/2$ yields BDF2, $\theta=1$ the midpoint rule, and $\theta=0$ a two-step midpoint scheme [2510.16860][2409.19481][2108.09339].

## 2. Nonlinear $G$-Stability and Energy-Dissipation Properties

The nonlinear and unconditional $G$-stability of DLN is central to its robust behavior on variable grids. Defining the $G$-norm for a pair $(u, v)$ by
\[
\| (u, v) \|^2_{G(\theta)} = \tfrac{1}{4}(1+\theta)\|u\|^2 + \tfrac{1}{4}(1-\theta)\|v\|^2,
\]
the method satisfies a telescoping energy identity:
\[
(y_{n,\alpha}, y_{n,\beta}) = \| (y_{n+1}, y_n)\|_{G(\theta)}^2 - \| (y_n, y_{n-1})\|_{G(\theta)}^2 + \|\sum_{\ell=0}^2 a_\ell^{(n)} y_{n-1+\ell} \|^2,
\]
for explicit $a_\ell^{(n)}$. For nonlinear dissipative problems, this structure yields a monotonic decay of the discrete energy, preventing nonphysical growth due to variation in step size, with or without viscosity [2510.16860][2509.19053][2407.19101][2001.08640][2108.09339]. In the linear test equation $y' = \lambda y$ with $\mathrm{Re}\,\lambda \le 0$, $G$-stability yields unconditional $A$-stability.

## 3. Consistency, Local Truncation Error, and Accuracy

The DLN method is globally second-order accurate for any time-grid with step ratios bounded away from zero. A Taylor expansion of the solution reveals that the local truncation error at each step $n$ is
\[
\tau_{n+1} = C(\theta, r_n)\,k_n^2\, y'''(\xi) + O(k_n^3),
\]
with $C(\theta, r_n)$ given explicitly, and $\xi \in (t_{n-1}, t_{n+1})$ [2510.16860][2409.19481][2108.09339]. Spatial discretizations, such as conforming FEM assemblages, combine to yield full error bounds of
\[
\|u_n^h - u(t_n)\| = O(k_{\max}^2 + h^{s+1}),
\]
under standard regularity assumptions and mild step-size moderation.

## 4. Refactorization and Efficient Implementation

Direct deployment of DLN requires managing non-trivial stage coefficients, but it admits exact algebraic refactorization as backward Euler plus linear time filtering. Each DLN step is realized as:
1. Linear combination of previous solutions (pre-filter),
2. Single backward Euler solve with effective time step and collocation,
3. Linear combination of new and old solutions (post-filter).

This structure dramatically reduces code complexity and enables embedding DLN into backward-Euler codebases with only minor augmentation [2306.02461][2108.09339][2409.19481]. All history dependence is contained in short recurrences, and storage requirements are minimal (two solution vectors per unknown plus filters).

## 5. Adaptive Time-Stepping and Minimum-Dissipation Criterion

A distinctive feature of modern DLN practice is robust adaptive control, notably via the minimum-dissipation criterion. At each step, the numerical dissipation (ND) and physical dissipation (PD; often associated to viscosity) are evaluated:
\[
\mathrm{ND} = \| w_{n,\alpha} \|^2 / \widehat k_n, \qquad \mathrm{PD} = \mu \| \nabla w_{n,\beta} \|^2.
\]
The ratio $\chi = \mathrm{ND}/\mathrm{PD}$ is compared to a tolerance $\delta$, and the step updated via:
\[
k_{n+1} =
\begin{cases}
\min(2k_n, k_{\max}), & \chi \le \delta \\
\max(\tfrac12 k_n, k_{\min}), & \chi > \delta
\end{cases}
\]
[2510.16860][2509.19053][2507.21392][2407.19101][2001.08640]. This ensures controlled energy dissipation and efficiently tracks dynamic stiffness. Alternative adaptivity strategies deploy embedded AB2-type local error estimation, but dissipation-based adaptivity aligns more directly with the $G$-stability framework for complex nonlinear systems.

## 6. Coupling With Mixed FEM and Application to Nonlinear Fourth-Order Systems

DLN integrators couple naturally with divergence-free preserving mixed FEM for nonlinear PDEs, such as the Allen–Cahn active fluid model or fourth-order active/dissipative flows. High-order operators (e.g., biharmonic) are handled via variable splitting (e.g., $w = \Delta u$), lowering the regularity requirements. Auxiliary, divergence-free variables (e.g., $\phi$ as Lagrange multiplier) and monolithic solvers are deployed at each time step [2510.16860][2509.19053][2507.21392]. Numerical experiments confirm that minimum-dissipation-adaptive DLN can yield dramatic reductions in total time steps (e.g., 500 instead of 20,000 for comparable solution accuracy at high Reynolds numbers) and remain stable for complex long-time nonlinear dynamics.

## 7. Practical Recommendations and Observations From Numerical Studies

Key practical guidelines include:
- Limiting the step-ratio $r_n \in [0.5, 2.0]$ stabilizes coefficient conditioning.
- Choices of $\theta$ influence damping and error: $\theta \in (0.3, 0.7)$ often yields the best compromise; $\theta=1/2$ matches BDF2 under uniform steps.
- Startup values require two initial states; Crank–Nicolson initialization is commonly used.
- Implementation overhead is minimal compared to implicit Euler, especially under refactorization.
- Tests on Navier–Stokes, Allen–Cahn, Stokes/Darcy, and active fluid models confirm unconditional nonlinear stability, second-order temporal accuracy, and significant efficiency gains under adaptivity, compared to BDF2 or trapezoidal integrators, especially in highly stiff or multiscale regimes [2510.16860][2509.19053][2507.21392][2409.19481][2001.08640][2108.09339].

---

**Summary Table: DLN One-Leg Two-Step Variable-Step Scheme**

| Step             | Formula                                                | Notes                                      |
|------------------|-------------------------------------------------------|--------------------------------------------|
| Recurrence       | $\sum_{\ell=0}^2 \alpha_\ell y_{n-1+\ell} = \widehat k_n f(t_{n,\beta},y_{n,\beta})$ | $\alpha$, $\widehat k_n$, $\beta$ as above |
| Coefficient Form | $\alpha_2 = \tfrac12(1+\theta),\ \alpha_1 = -\theta,\ \alpha_0 = \tfrac12(\theta-1)$ | $\theta\in[0,1]$                          |
| Adaptivity       | If $\chi\le\delta,\ k_{n+1}=\min(2 k_n, k_{\max})$, else $k_{n+1}=\max(\tfrac12 k_n, k_{\min})$ | Based on ND/PD ratio                      |

This synthesis is directly grounded in the key recent and foundational references on the DLN integrator family [2510.16860][2509.19053][2507.21392][2409.19481][2306.02461][2407.19101][2007.03801][2001.08640][2108.09339].

Source: https://www.emergentmind.com/topics/variable-step-dahlquist-liniger-nevanlinna-dln-time-integrator-3be2d178-6f64-4a77-9fd6-3877e2b9fc4c