LinOSS-IMEX: Implicit-Explicit Time Integration

• LinOSS-IMEX is a time integration framework that partitions stiff and nonstiff dynamics via operator splitting and hybrid Runge–Kutta methods.
• The scheme employs selective implicit treatment of stiff components and explicit updates for nonstiff ones, enhancing stability and computational efficiency.
• It underpins applications in financial option pricing, multiscale fluid–structure interactions, and diffusion-dominated processes, demonstrating significant performance gains.

Implicit-Explicit Time Integration (LinOSS-IMEX) schemes are a class of time discretization strategies for stiff systems of partial differential equations (PDEs) or ordinary differential equations (ODEs), particularly suited to problems exhibiting multiple spatial or temporal scales, such as nonlinear parabolic PDEs in finance, multiscale fluid–structure interactions, and advection-diffusion–reaction equations. LinOSS-IMEX schemes combine operator splitting of linear/nonlinear or stiff/nonstiff terms (the “LinOSS” or linear-operator splitting paradigm) with advanced implicit-explicit integration techniques, generally of Runge–Kutta or general linear method (GLM) type, to maximize both stability and computational efficiency. The methodology is characterized by selective, stage-wise implicit treatment of stiff (often diffusive or acoustic) linear components and explicit, high-order accurate, nonstiff handling of advective, nonlinear, or source terms.

1. Formulation and Operator Splitting Principles

The LinOSS-IMEX framework begins with the identification and decomposition of a semi-discrete system derived from a spatial discretization (e.g., finite volume, finite element, spectral) of a PDE. The canonical structure is

$\frac{dU}{dt} = F(U) + S(U)$

where $F(U)$ denotes the non-stiff (explicit) part—commonly advective, nonlinear, or reactive—and $S(U)$ the stiff (implicit) part, typically linear diffusion or fast wave propagation (López-Salas et al., 2 Sep 2024).

A representative example is the one-dimensional parabolic PDE (with possible nonlinearities) in option pricing: $$\frac{\partial u}{\partial t} + a(x,t)\frac{\partial u}{\partial x} + b(x,t)\frac{\partial^2 u}{\partial x^2} + H(x,t,u,\partial_x u) = 0$$ which is recast in conservative form, discretized in space by a finite-volume scheme (explicit advection, implicit diffusion), leading to the split ODE system above (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024).

IMEX time integrators then advance $U$ by treating $F(U)$ explicitly (Runge–Kutta, multistep, or GLM stages) and $S(U)$ implicitly (often via diagonally implicit RK, BDF, or related solvers) (Albi et al., 2013, Zhang et al., 2013). This splitting is fundamental to achieving enlarged stability regions (CFL relaxation for stiff terms) without the complexity or computational cost of fully implicit solves for the entire system.

2. IMEX Time Integration Schemes

The LinOSS-IMEX approach employs time integrators defined by a pair of Butcher tableaus—one explicit for non-stiff operators, one implicit for stiff operators—constructed to reach at least second order globally. For example, the L-stable, second order IMEX-SSP2(2,2,2) scheme is characterized by

Explicit (ERK) tableau:

$\begin{array}{c|cc} 0 & 0 & 0 \ 1 & 1 & 0 \ \hline & \tfrac12 & \tfrac12 \end{array}$

Implicit (DIRK) tableau:

$\begin{array}{c|cc} \gamma & \gamma & 0 \ 1-\gamma & 1-2\gamma & \gamma \ \hline & \tfrac12 & \tfrac12 \end{array} \qquad \gamma = 1-\frac{1}{\sqrt{2}}$

with stages given, for each time interval $[t^n, t^{n+1}]$, by a sequence of implicit and explicit updates (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024). The IMEX methodology generalizes to higher orders via multi-stage ARS (Ascher-Ruuth-Spiteri) (Gopinath et al., 2022), BPR, or GLM-based schemes (Zhang et al., 2013).

3. Stage Equations, Stability, and CFL Conditions

For a $s$-stage LinOSS-IMEX-RK scheme:

• Non-stiff (explicit) terms are advanced via forward-explicit (Heun-type, RK2, or higher order) updates;
• Stiff (implicit) terms are advanced by inverting diagonal or lower-triangular systems per stage (often only diffusion or linear fast waves).

Typical update for two-stage IMEX-SSP2(2,2,2): $$\begin{aligned} U^{(1)} &= U^n + \gamma\,\Delta t\,S(U^{(1)}) \ U^{(2)} &= U^n -\Delta t\,F(U^{(1)}) + (1-2\gamma)\,\Delta t\,S(U^{(1)}) + \gamma\,\Delta t\,S(U^{(2)}) \ U^{n+1} &= U^n - \frac{\Delta t}{2} \big[F(U^{(1)}) + F(U^{(2)})\big] + \frac{\Delta t}{2} \big[S(U^{(1)}) + S(U^{(2)})\big] \end{aligned}$$ which is provably second-order in time (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024).

LinOSS-IMEX schemes only require the explicit part (advection/convection) to satisfy a hyperbolic-type CFL

$$\alpha\,\frac{\Delta t}{\Delta x} \le \mathrm{CFL}_{\rm adv}\sim0.5$$

with $\alpha$ the max explicit advection wave speed. The parabolic (diffusion) CFL of explicit schemes, $\Delta t \leq C (\Delta x)^2$, is absent due to the implicit treatment of $S$ (López-Salas et al., 2 Sep 2024). The implicit tableau is chosen L-stable to encompass the stiff eigenvalues. This structure extends to 2D and systems with mixed derivatives (López-Salas et al., 2 Sep 2024), as well as to various application domains.

4. Extensions, Order Reduction, and High-Order Strategies

LinOSS-IMEX supports:

• High-order in time via multi-stage IMEX-RK schemes (e.g., ARS343 for order 3, KC664 for order 4) (Gopinath et al., 2022).
• High-order in space by replacing FV linear reconstructions with CWENO, DG, or compact-difference stencils (Singh et al., 2019).

A noted phenomenon is order reduction of high-order schemes in problems involving DAEs or extreme stiffness (large diffusion, small relaxation), notably in higher-order IMEX-RKs in Boussinesq convection and Goldstein–Taylor systems (Gopinath et al., 2022, Albi et al., 2013). For problems with DAEs, globally stiffly accurate (GSA) or special DAE-aware IMEX schemes are preferred to avoid loss of convergence order (Gopinath et al., 2022).

IMEX general linear methods (IMEX-GLMs), particularly DIMSIM-type with stage order $q=p$, avoid the mixed-term order reduction of classical IMEX-RK, retaining full convergence order even for stiff–nonstiff partitioning (Zhang et al., 2013).

5. Applications and Numerical Performance

LinOSS-IMEX schemes are broadly deployed in:

• Nonlinear parabolic PDEs in option pricing, enabling accurate solution of price and Greeks with non-regular terminal payoffs, while allowing parabolic time steps proportional to $\Delta x$ rather than $(\Delta x)^2$ (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024).
• High-Re, stiff-diffusion or DAE systems, such as pseudo-spectral Boussinesq convection, where third- and fourth-order IMEX-RK schemes (ARS343, KC664) outperform classical CNAB2 integrators in both speed and accuracy at their maximum stable $\Delta t$ (Gopinath et al., 2022).
• ADR equations, where the IMEX midpoint/RK2 with compact-difference stencils achieves large regions of neutral stability, minimal phase error, and suppression of spurious q-waves (Singh et al., 2019).
• Coupled free flow–porous media models with nonlinear interface (Lions condition), exploiting SAV-based IMEX schemes for provable unconditional stability and linear, decoupled algebraic solves (Wang et al., 18 May 2024).
• Multiscale hyperbolic systems, where LinOSS-IMEX (often with TVD or MOOD mechanisms) enables scale-independent CFL, non-oscillatory solutions, and robust detection/limiting (Michel-Dansac et al., 7 Jan 2025).
• Industrial compressible flow simulations, using cell-wise explicit–implicit hybridization (AION/LinOSS-IMEX) to blend Heun and Crank–Nicolson advances optimally in space (Muscat et al., 2019).

In all applications, LinOSS-IMEX methods yield substantial gains in allowable time steps, CPU time, and solution regularity under stiff or heterogeneous conditions.

6. Implementation and Algorithmic Details

Efficient LinOSS-IMEX integrators involve:

• Assembly of explicit and implicit operator splits per time stage;
• Piecewise linear or higher-order reconstruction, often with non-oscillatory TVD or WENO limiters for spatial derivatives;
• For each time step, solution of two (2nd order) or more (higher order) implicit diffusion (or stiff linear) algebraic systems, typically sparse and efficiently solvable by factorization or iterative methods;
• Weighted combination of explicit/implicit stage outputs to update the solution;
• Implicit enforcement of boundary conditions in the solution of the diffusion or pressure operator;

The methods are compatible with a variety of ODE solvers (Newton/fixed-point for nonlinear diffusion, direct or Krylov solvers for linear systems) and can exploit precomputed matrix factorizations for constant-coefficient implicit operators (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024, Albi et al., 2013).

For problems with severe scale separation (e.g., low Mach flows), LinOSS-IMEX–MOOD defers to stricter TVD variants in troubled cells, reverting to first-order to guarantee non-oscillatory behavior, while maintaining high order in smooth regions (Michel-Dansac et al., 7 Jan 2025).

7. Theoretical Properties and Numerical Evidence

LinOSS-IMEX schemes possess:

• Rigorous second-order (or higher) global accuracy, with error estimates in $L^1$, $L^\infty$, and energy-based norms under broad regularity assumptions (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024, Wang et al., 18 May 2024).
• Unconditional (often CFL-free) stability with respect to stiff linear terms, owing to L-stability of the implicit tableau (López-Salas et al., 2 Sep 2024, Wang et al., 18 May 2024).
• Uniform convergence and asymptotic preservation in singular-perturbation regimes, e.g., in Goldstein–Taylor diffusion, where the discrete scheme preserves the correct limit as $\varepsilon \rightarrow 0$ (Albi et al., 2013).
• Demonstrated computational superiority over fully explicit or fully implicit schemes in both benchmark and real-world test cases; performance gains up to $10^4\times$ are reported in large-N, diffusion-dominated settings (López-Salas et al., 2 Sep 2024, López-Salas et al., 2 Sep 2024, Gopinath et al., 2022).
• Robustness to non-regular initial or boundary data, and preservation of monotonicity or TVD properties with appropriate limiting (López-Salas et al., 2 Sep 2024, Wang et al., 18 May 2024, Michel-Dansac et al., 7 Jan 2025).

A plausible implication, given the breadth of tested problems, is that LinOSS-IMEX methodology is broadly transferrable to other stiff systems, provided proper operator splitting and tableau selection.

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References

• “IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing” (López-Salas et al., 2 Sep 2024)
• “Second order finite volume IMEX Runge-Kutta schemes for two dimensional parabolic PDEs in finance” (López-Salas et al., 2 Sep 2024)
• “An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus” (Gopinath et al., 2022)
• “Implicit-explicit-compact methods for advection diffusion reaction equations” (Singh et al., 2019)
• “Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model” (Albi et al., 2013)
• “A class of new linear, efficient and high-order implicit-explicit methods for the coupled free flow-porous media system based on nonlinear Lions interface condition” (Wang et al., 18 May 2024)
• “TVD-MOOD schemes based on implicit-explicit time integration” (Michel-Dansac et al., 7 Jan 2025)
• “A coupled implicit-explicit time integration method for compressible unsteady flows” (Muscat et al., 2019)
• “Partitioned and implicit-explicit general linear methods for ordinary differential equations” (Zhang et al., 2013)