A commutator and equilibrium-projection obstruction for stiff splitting

Definitions and mathematical statement

The original item is Margot Gerritsen's participant contribution, jointly with Rami Younis, about efficient reservoir flow and transport simulation. It discusses miscible gas injection, in-situ combustion or fire-flooding, multiscale flow and energy equations, strongly nonlinear weakly hyperbolic component transport, stiff reaction, Euler--Lagrangian methods, and splitting. It poses no explicit mathematical conjecture. The phrase "which either parabolic in character" is incomplete already in the primary document according to the aid; no missing equation or adjective is guessed here.

The supplied auxiliary diagnostic concerns the linear periodic system \[ U_t+A U_x=\varepsilon^{-1}RU,\qquad\varepsilon>0, \] where $A$ is a constant real symmetric matrix and $R$ is constant real symmetric negative semidefinite. On $L^2$, transport $T_h=e^{-hA\partial_x}$ is unitary and reaction $E_h=e^{hR/\varepsilon}$ is contractive. For $U_0\in H^1$, the aid's reaction-last Lie local-error bound is \[ \|E_hT_hU_0-e^{h(-A\partial_x+R/\varepsilon)}U_0\|_2 \le\frac{h^2}{2\varepsilon}\|[R,A]\|\,\|\partial_xU_0\|_2, \qquad[R,A]=RA-AR. \] For a fixed Fourier wavenumber $\kappa$, the split generators are $B=-i\kappa A$, $C=R/\varepsilon$. Lie and Strang local expansions involve $[C,B]$, and respectively the nested commutators $[B,[B,C]]$ and $[C,[C,B]]$, exposing $\varepsilon^{-1}$ and $\varepsilon^{-2}$ factors. Fixed-$\varepsilon$ formal order therefore does not establish stiff-uniform order.

Let $P$ be the orthogonal projection onto $\ker R$. The equilibrium-projection criterion asserted by the aid is \[ P e^{-izA}P=e^{-izPAP}P\quad\text{for every sufficiently small real }z \quad\Longleftrightarrow\quad[P,A]=0. \] The left side is transport followed by equilibrium projection on already equilibrated data; the right side is the exact reduced transport. Symmetry of $A$ is essential for the stated equivalence: the second-order defect contains $PA(I-P)AP=((I-P)AP)^*((I-P)AP)$.

An exact two-component witness is \[ A=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad R=\begin{pmatrix}0&0\\0&-1\end{pmatrix},\qquad P=\begin{pmatrix}1&0\\0&0\end{pmatrix}. \] On the Fourier mode $\kappa=1$, take equilibrated initial amplitude $e_1$. The reaction-half/transport/reaction-half Strang matrix is \[ S_{h,\varepsilon}=e^{hR/(2\varepsilon)}e^{-ihA}e^{hR/(2\varepsilon)}. \] For fixed $h>0$, its stiff limit is $P e^{-ihA}P=(\cos h)P$, while $PAP=0$, so the exact reduced slow amplitude remains $e_1$. After $N=T/h$ steps, with $N$ an integer and $h\downarrow0$, the limiting error amplitude is \[ 1-(\cos h)^{T/h}=\frac T2h+O(h^2). \] The limit order is $\varepsilon\to0$ at fixed $h,N$, followed by $h\to0$ at fixed $T$. It displays first-order global slow error in that stiff limit despite the nominal fixed-$\varepsilon$ second order of Strang splitting. It is not a claim that every splitting method or reservoir system has this error, nor a solution of the participant's nonlinear multiphase modelling programme.

Short English statement

The source is a joint research-interest statement. The aid supplies a linear stiff-splitting diagnostic: transport must preserve the reaction-equilibrium subspace for the projected step to match exact reduced transport. An explicit two-component example otherwise has first-order global slow error in the stiff limit.

Sources

• Local registry, permanent UnsolvedMath ID $20002461$, exact public code catalogue entry; canonical location aim-pdes-notes.json, notes[72].
• Margot Gerritsen, joint contribution with Rami Younis, A.8, AIM Stiff Sources and Numerical Methods for Conservation Laws, version 1 April 2005; supplied reference https://aimath.org/WWN/balancelaws/balancelaws.pdf. The aid reports the incomplete parabolic-character phrase is already in the primary PDF. No PDF was accessed here.
• Supplied research attempt 1: context_only, run status not_a_problem, rigor label proved, low novelty confidence; registry status open, classification OPEN-TRIAGE, created and updated 14 August 2026. No original remarks, literature entry, or primary reference for the auxiliary commutator and projection results is supplied.
• The research aid explicitly says it is machine generated, not peer reviewed, and requires independent verification before citation.