Additive (IMEX) methods (IMEX) — 7 schemes

An additive split separates a part that can be evaluated explicitly from a stiff part that needs an implicit solve. The explicit part still restricts the step. A split is useful only when it reflects the actual equations and makes the solve cheaper.

For the split right-hand side $y' = f_E(t, y) + f_I(t, y)$: the smooth (non-stiff) part explicit, the stiff part implicit. The split is read by capability through rhs.resolve (see the numerical record): an IMEXRHS (or a structured RHS providing explicit/implicit) activates the split; an ordinary RHS callable degenerates to the pure-implicit limit of the scheme, so the standard integrator call convention stays valid.

IMEX Euler (IMEXEuler)

One step: explicit Euler on $f_E$ at the start point, one JFNK solve for the implicit endpoint:

$$y^{n+1} = y^n + \mathrm{dt}\, f_E(t^n, y^n) + \mathrm{dt}\, f_I(t^{n+1}, y^{n+1}).$$

Order 1; L-stable on the implicit half; the pure-implicit limit is backward Euler. One evaluation + one solve per step.

ARK3(2)4L[2]SA (ARK324L2SA)

Kennedy-Carpenter's additive RK pair: 4 stages, implicit half order 3, explicit half order 2, L[2]-stable. The implicit half ($a_{\text{imp}}, b, d$) is identical in $a/b/c$ to the registered ESDIRK3(2)4L[2]SA; the explicit half is the matching $a_{\text{exp}}$ (full matrices in ark.getARKTableau), and both halves share the single embedded $d$ vector. Stiffly accurate on the implicit side only — the explicit half is not stiffly accurate, so priorStep is refused (reuse would corrupt the explicit branch). Cost: one JFNK solve + $s$ explicit evaluations per step.

ARK4(3)6L[2]SA (ARK436L2SA)

Kennedy-Carpenter's 6-stage pair: implicit half order 4, explicit half order 3, L[2]-stable, carried in schemes.coefficients.additive in exact rational form ($c = (0, 1/2, 83/250, 31/50, 17/20, 1)$). Note the implicit half is a different 6-stage ESDIRK design from the standalone ESDIRK4(3)6L[2]SA (different nodes: $83/250, 31/50, 17/20$ vs. $(2-\sqrt2)/4, 5/8, 26/25$), so its coefficients are carried in full rather than by reference.

Stability slices. The additive pair's stability is two-parameter (implicit eigenvalue scale × explicit eigenvalue scale); the slices are measured in tests/test_ark.py (see the numerical record): the L[2] implicit half damps its own stiff modes to $O(z^{-2})$ while the explicit half must stay inside its own region — the practical CFL is set by the smooth part, which is the point of the split.

IMEX linear multistep — SBDF2, SBDF3, CNAB2

The multistep additive family (drivers.history.imex, the numerical record): a BDF or trapezoidal implicit backbone for the stiff part plus explicit endpoint extrapolation for the smooth part.

SBDF$p$ ($p = 2, 3$): the BDF$p$ backbone with the BDF derivative weight $\beta$ scaling the whole endpoint derivative —

$$y^{k+1} = \sum_j c_j\, y^{k-j} + \beta\, \mathrm{dt}\, \big[ f_I(t^{k+1}, y^{k+1}) + E(f_E) \big],$$

where $E$ is the $p$-point Lagrange polynomial through the known states, extrapolated to $t^{k+1}$:

$$\text{SBDF2: } E(f_E) = 2 f_E^k - f_E^{k-1}, \qquad \text{SBDF3: } E(f_E) = 3 f_E^k - 3 f_E^{k-1} + f_E^{k-2}.$$

Combined order $p$; order defects $O(h^3)$ (SBDF2), $O(h^4)$ (SBDF3), verified by Taylor expansion in tests/test_imexmultistep.py. Dropping $\beta$ on the explicit part amplifies it by $1/\beta$ and collapses the defect to $O(h)$ — measured, and pinned as a regression guard. Pure-implicit limit: BDF$p$ (SBDF2 A- and L-stable, SBDF3 A($\alpha$) 86.03°); pure-explicit limit: the matching zero-stable explicit $p$-step method.

CNAB2 (order 2): trapezoidal on the implicit part + AB2 increment on the explicit part:

$$y^{k+1} = y^k + \tfrac{\mathrm{dt}}{2}\big[f_I(t^k, y^k) + f_I(t^{k+1}, y^{k+1})\big] + \mathrm{dt}\big[\tfrac{3}{2} f_E^k - \tfrac{1}{2} f_E^{k-1}\big].$$

Pure-implicit limit: the trapezoidal rule (A-stable, not L); pure-explicit limit: AB2.

History and cold start follow the BDF family exactly (state-snapshot StepHistory, DP5 cold start, priorStep refused). Diagnostics: the step's one solve plus the non-solve evaluations (the explicit-part evaluations, the CNAB2 implicit start-point evaluation, the endpoint history evaluation) are counted into rhs_evaluations, so the work-unit invariant covers the full step cost. Per-step TVD: measured to CFL 1.5 with the rest of the multistep family (see the numerical record).

Crank–Nicolson–Leapfrog (cnlf2)

One implicit solve per step closes y[n+1] = y[n-1] + h*(I[n-1] + I[n+1]) + 2*h*E[n]. Thread result.history through history= or use the ordinary solve runner. The first step is IMEX Euler; a finite startup defect preserves second-order global convergence. Repeated cold calls remain first order. Changed step size, time discontinuity, layout or particle IDs reset history before consumption; use restart_history=True after changing RHS parameters.

The pure implicit two-step limit is A-stable, with a parasitic mode and no stiff damping. At hλ=−100, y0=1, y2=−99/101. The explicit leapfrog part supports oscillatory spectra within its CFL interval and is unstable for negative-real decay. Neither A-stability nor the historical cold-start TVD classifier implies monotonicity of the warm split. This is a fixed-step method without an estimator. Input/history ownership, failed-solve rollback, parameter gradients and portable tensor restart are covered by the backend record.

When to use

• IMEX Euler: the cheapest split; order 1 is often enough when the stiff part is a linear damping term.
• ARK3/4: the workhorse splits — L[2] damping of the stiff part at order 3/4 with no cost on the smooth part.
• SBDF2/3, CNAB2: multistep splits when per-step cost must stay near one solve; SBDF2 is the A/L-stable one.

References

• Ascher, Ruuth, Rapada, Implicit-explicit Runge-Kutta methods for ordinary differential equations, Proc. R. Soc. Lond. A 459 (2003) 427-443 — the ARK construction.
• Kennedy & Carpenter (2001/2002) — the L[2]SA designs (as shipped in SUNDIALS ARKODE v7.9.0).
• Hairer & Wanner, Solving ODEs II (1996) — IMEX and BDF theory.

Tests: tests/test_imex.py (IMEX Euler), tests/test_ark.py (ARK order, stability slices, the pure-implicit limit), tests/test_imexmultistep.py (SBDF/CNAB orders, defects, the $\beta$ regression guard), tests/test_rhs.py (the capability-based split resolution).

The numerical record retains derivations, source pins and measured property gates.