For the optional splitting branch of the finite-dimensional coupled model, suppose $a,b\in C^3(U,{\mathbb{R}}^d)$ define the additive autonomous ODE of Additive autonomous ODE partitions, their exact subflows use the Lie and Strang compositions of Lie and Strang compositions of exact split flows, and all exact and numerical paths lie with positive margin in a common compact convex tube on $[0,T]$. Initialize both schemes at the exact initial value $y_0=y(0)$. Under the remaining Lipschitz, regularity, and step-size hypotheses of Lie and Strang splitting have global orders one and two, with separate subflow errors, the exact Lie and Strang schemes have global errors $O(h)$ and $O(h^2)$ respectively. Approximate subflows retain these orders only under the separate per-subflow error budgets and composition stability condition stated in that theorem. This corollary makes no claim for unbounded operators or general moving-domain PDEs.

Let $I_1,I_2$ and the common trace space $\mathrm{\Lambda }$ be finite-dimensional real inner-product spaces, with $\dim \mathrm{\Lambda }\ge 1$, and set $X_i:=I_i\oplus \mathrm{\Lambda }$. The trace map is ${\gamma }_i(x_i,{\lambda }_i)={\lambda }_i$. Let $M_i$ and $A_i$ be self-adjoint positive-definite operators on $X_i$, let $\mathrm{\Delta }t>0$, and define the local backward-Euler block $B_i:=\mathrm{\Delta }{t}^{-1}{M}_i+A_i$. In coordinates $I_i\oplus \mathrm{\Lambda }$, write

$B_i=\left(\begin{array}{cc}{B}_{i,00}& B_{i,0\mathrm{\Gamma }}\\ B_{i,\mathrm{\Gamma }0}& B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}\end{array}\right),g_i=(g_{i,0},g_{i,\mathrm{\Gamma }})\mathrm{.}$

For a prescribed local trace ${\lambda }_i$, solve the interior equation $B_{i,00}{x}_i+B_{i,0\mathrm{\Gamma }}{\lambda }_i=g_{i,0}$ and define the outward algebraic interface flux $q_i:=B_{i,\mathrm{\Gamma }0}{x}_i+B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}{\lambda }_i-g_{i,\mathrm{\Gamma }}$. Eliminating $x_i$ gives $q_i=S_i{\lambda }_i-f_i$, where $S_i:=B_{i,\mathrm{\Gamma }\mathrm{\Gamma }}-B_{i,\mathrm{\Gamma }0}{B}_{i,00}^{-1}{B}_{i,0\mathrm{\Gamma }}$ and $f_i:=g_{i,\mathrm{\Gamma }}-B_{i,\mathrm{\Gamma }0}{B}_{i,00}^{-1}{g}_{i,0}$. The sign convention is outward flux on each subdomain.

The monolithic interface equations are trace continuity ${\lambda }_1={\lambda }_2=\lambda$ and flux balance $q_1+q_2=0$, equivalently $(S_1+S_2)\lambda =f_1+f_2$. These finite-dimensional matrices represent a linear diffusion-transmission step; no nonlinear fluid-structure well-posedness is asserted.

Fix an integer $d\ge 1$, a time interval $[0,T]$ with $T>0$, and a nonempty bounded open reference domain $\widehat{\mathrm{\Omega }}\subset {\mathbb{R}}^d$ with compact Jordan closure; Jordan measurability means equality of finite-rectangle inner and outer contents (Jordan inner and outer content and Jordan measurable bounded sets in ${\mathbb{R}}^m$). In dimension $d=1$, require $\widehat{\mathrm{\Omega }}$ to be a finite union of bounded open intervals with pairwise disjoint closures. In dimensions $d\ge 2$, require a finite piecewise-$C^1$ boundary presentation: a finite family of compact boundary faces $S_1,\dots ,S_q$ and a compact edge set $E$ such that the faces cover $\mathrm{\partial }\widehat{\mathrm{\Omega }}$, each face lies in a regular $C^1$ hypersurface patch, $S_i\cap E$ is surface-null, all face overlaps lie in $E$, and off $E$ the boundary is locally one $C^1$ graph with $\widehat{\mathrm{\Omega }}$ on one side. The faces carry that side's outward unit normal.

An ALE moving-domain map is a map $A:[0,T]\times \overline{\widehat{\mathrm{\Omega }}}\to {\mathbb{R}}^d$ whose space-time restriction extends to a $C^2$ map on a neighborhood of this set, and for each $t$ the map $A_t(\xi ):=A(t,\xi )$ is an orientation-preserving diffeomorphism on neighborhoods of the closures. Write $\mathrm{\Omega }(t):=A_t(\widehat{\mathrm{\Omega }})$ and $J(\xi ,t):=\det D_{\xi }{A}_t(\xi )>0$. Define the mesh velocity in physical coordinates by $w_m(x,t):={\mathrm{\partial }}_t{A}_t(A_t^{-1}(x))$. In $d=1$ the moving boundary consists of the transported interval endpoints. In $d\ge 2$, require the transported faces under $A_t$ to satisfy the same finite piecewise-$C^1$ boundary-presentation conditions; the diffeomorphism carries the reference faces and edge set to that presentation, with the transported outward normal.

This definition fixes the regularity and sign conventions for the continuous ALE identities. It does not assert a discrete approximation property, a nonintersection theorem for a computed mesh, or well-posedness of a coupled PDE.

Fix $N\ge 1$, a reference cell width $\mathrm{\Delta }\xi >0$, a time step $\mathrm{\Delta }t>0$, and a physical period length $L>0$. Face positions $x_{i+1\mathrm{/}2}^n$ and face velocities $w_{i+1\mathrm{/}2}^n$ are indexed periodically by $i\in \mathbb{Z}$, with $x_{i+N+1\mathrm{/}2}^n=x_{i+1\mathrm{/}2}^n+L$ and $w_{i+N+1\mathrm{/}2}^n=w_{i+1\mathrm{/}2}^n$. The physical cell $i$ is $[x_{i-1\mathrm{/}2}^n,x_{i+1\mathrm{/}2}^n]$ and its dimensionless Jacobian is $J_i^n=(x_{i+1\mathrm{/}2}^n-x_{i-1\mathrm{/}2}^n)\mathrm{/}\mathrm{\Delta }\xi$; require $J_i^n>0$.

For a constant physical advection speed $a\in \mathbb{R}$, the relative face speed is $c_{i+1\mathrm{/}2}^n:=a-w_{i+1\mathrm{/}2}^n$. Write $c^{+}:=\max (c,0)$ and $c^{-}:=\min (c,0)$. The prescribed face move is $x_{i+1\mathrm{/}2}^{n+1}=x_{i+1\mathrm{/}2}^n+\mathrm{\Delta }t{w}_{i+1\mathrm{/}2}^n$. The cell average ${\bar{u}}_i^n$ is the average over the corresponding physical cell.

For positive scalars $m_s$ and $m_f$, call $m_s$ the structure mass and $m_f$ the fluid added mass in the scalar toy added-mass recurrence. Given a scalar initial velocity increment $\delta v_0$, define successive increments by

$m_s\delta v_{n+1}=-m_f\delta v_n(n\ge 0)\mathrm{.}$

This is a stipulated one-mode sequential update used only to expose its amplification factor. It is not derived from, and makes no stability claim about, a general fluid-structure discretization.

Inputs: The periodic grid and relative speeds of Periodic one-dimensional moving grid, cell averages ${\bar{u}}_i^n\in \mathbb{R}$, a constant physical speed $a$, face velocities $w_{i+1\mathrm{/}2}^n$, $\mathrm{\Delta }t>0$, and $\mathrm{\Delta }\xi >0$.

Instructions:

1. Without modifying the input mesh or cell averages, compute trial face positions $x_{i+1\mathrm{/}2}^{n+1}=x_{i+1\mathrm{/}2}^n+\mathrm{\Delta }t{w}_{i+1\mathrm{/}2}^n$ and trial Jacobians $J_i^{n+1}=(x_{i+1\mathrm{/}2}^{n+1}-x_{i-1\mathrm{/}2}^{n+1})\mathrm{/}\mathrm{\Delta }\xi$.

2. If any $J_i^{n+1}\le 0$, discard the trial data and return the status mesh_crossing and no updated state, leaving the input mesh and cell averages unchanged.

3. Otherwise, for every face set $c_{i+1\mathrm{/}2}^n=a-w_{i+1\mathrm{/}2}^n$ and ${\widehat{F}}_{i+1\mathrm{/}2}^n=(c_{i+1\mathrm{/}2}^n{)}^{+}{\bar{u}}_i^n+(c_{i+1\mathrm{/}2}^n{)}^{-}{\bar{u}}_{i+1}^n$, with periodic indices and $c^{+}=\max (c,0)$, $c^{-}=\min (c,0)$.

4. Accept the trial mesh and return the status success and the unique cell averages defined by $J_i^{n+1}{\bar{u}}_i^{n+1}=J_i^n{\bar{u}}_i^n-(\mathrm{\Delta }t\mathrm{/}\mathrm{\Delta }\xi )({\widehat{F}}_{i+1\mathrm{/}2}^n-{\widehat{F}}_{i-1\mathrm{/}2}^n)$.

All state updates occur only after every new cell Jacobian has passed the positivity check.

For the ALE moving-domain map $A_t$ of ALE moving-domain map and mesh velocity, let $F(\xi ,t):=D_{\xi }{A}_t(\xi )$ and $J(\xi ,t):=\det F(\xi ,t)$. Then

${\mathrm{\partial }}_{t}J(\xi ,t)=J(\xi ,t){\mathrm{div}}_x{w}_m(A_t(\xi ),t)\mathrm{.}$

This is the continuous geometric conservation identity. It makes no discrete or time-integration claim.

For the face motion of Periodic one-dimensional moving grid, every cell satisfies

$J_i^{n+1}=J_i^n+\frac{\mathrm{\Delta }t}{\mathrm{\Delta }\xi }(w_{i+1\mathrm{/}2}^n-w_{i-1\mathrm{/}2}^n)\mathrm{.}$

Let $B=\left(\begin{array}{cc}K& C\\ C^{\ast }& D\end{array}\right)$ be a self-adjoint positive-definite operator on a finite-dimensional real inner-product direct sum $I\oplus \mathrm{\Lambda }$. Then $K$ is positive definite, $S:=D-C^{\ast }{K}^{-1}C$ is positive definite on $\mathrm{\Lambda }$, and for every $z\in \mathrm{\Lambda }$,

$⟨Sz,z⟩={\min }_{x\in I}⟨B(x,z),(x,z)⟩\mathrm{.}$

In particular, the local Schur operators $S_i$ in Finite-dimensional backward-Euler diffusion transmission model are positive definite.

Assume the finite-dimensional diffusion-transmission model of Finite-dimensional backward-Euler diffusion transmission model. Let $V:=\{(v_1,v_2)\in X_1\oplus X_2:{\gamma }_1{v}_1={\gamma }_2{v}_2\}$ be the common-trace space, take $u^n\in V$, and let $F_i^{n+1}\in X_i$. In the block notation of the definition, set the local right-hand side to $g_i:=\mathrm{\Delta }{t}^{-1}{M}_i{u}_i^n+F_i^{n+1}$; assemble the local equations with common trace and the outward algebraic fluxes $q_i$ from that definition. There is a unique $u^{n+1}\in V$ satisfying

${\sum }_{i=1}^2(⟨M_i\frac{u_i^{n+1}-u_i^n}{\mathrm{\Delta }t},v_i⟩+⟨A_i{u}_i^{n+1},v_i⟩)={\sum }_{i=1}^2⟨F_i^{n+1},v_i⟩(v\in V)\mathrm{.}$

It obeys the exact identity

$\frac{1}{2}(\mathrm{\parallel }{u}^{n+1}{\mathrm{\parallel }}_M^2-\mathrm{\parallel }{u}^n{\mathrm{\parallel }}_M^2+\mathrm{\parallel }{u}^{n+1}-u^n{\mathrm{\parallel }}_M^2)+\mathrm{\Delta }t\mathrm{\parallel }{u}^{n+1}{\mathrm{\parallel }}_A^2=\mathrm{\Delta }t{\sum }_{i=1}^2⟨F_i^{n+1},u_i^{n+1}⟩,$

where $\mathrm{\parallel }v{\mathrm{\parallel }}_M^2:={\sum }_i⟨M_i{v}_i,v_i⟩$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_A^2:={\sum }_i⟨A_i{v}_i,v_i⟩$. In the unforced case the monolithic backward-Euler energy is nonincreasing.

Inputs: Positive-definite real Schur operators $S_1,S_2$ on the nonzero finite-dimensional trace space $\mathrm{\Lambda }$, loads $f_1,f_2\in \mathrm{\Lambda }$, a parameter $\alpha >0$, initial incoming datum $r_1^0\in \mathrm{\Lambda }$, tolerance $\epsilon >0$, and integer pass budget $M\ge 1$.

All vector norms on $\mathrm{\Lambda }$ and operator norms on its endomorphisms are induced by its specified real inner product. Set $R_i:=(\alpha I-S_i)(\alpha I+S_i{)}^{-1}$, $q_i:=\mathrm{\parallel }{R}_i\mathrm{\parallel }$, and $q:=q_1{q}_2$. For each pass $m=0,\dots ,M-1$, perform in order

$(S_1+\alpha I){\lambda }_1^{m+1}=r_1^m+f_1,r_2^{m+1}=(\alpha I-S_1){\lambda }_1^{m+1}+f_1;(S_2+\alpha I){\lambda }_2^{m+1}=r_2^{m+1}+f_2,r_1^{m+1}=(\alpha I-S_2){\lambda }_2^{m+1}+f_2\mathrm{.}$

If $\mathrm{\parallel }{r}_1^{m+1}-r_1^m\mathrm{\parallel }\le (1-q)\epsilon \mathrm{/}\mathrm{\parallel }(\alpha I+S_1{)}^{-1}\mathrm{\parallel }$, return converged with trace ${\lambda }_1^{m+1}$ and pass count $m+1$. If all $M$ passes are used without this test succeeding, return budget_exhausted with the last trace ${\lambda }_1^M$ and pass count $M$. This procedure states the finite stop predicate but makes no claim about its error certificate.

Let $S:=S_1+S_2$, let ${\lambda }_{\ast }$ solve $S{\lambda }_{\ast }=f_1+f_2$, and let $\tilde{\lambda }\in \mathrm{\Lambda }$ have residual $\rho :=S\tilde{\lambda }-(f_1+f_2)$. All trace-vector and operator norms below are induced by the given real inner product on $\mathrm{\Lambda }$. Then

$\mathrm{\parallel }\tilde{\lambda }-{\lambda }_{\ast }\mathrm{\parallel }\le \mathrm{\parallel }{S}^{-1}\mathrm{\parallel }\mathrm{\parallel }\rho \mathrm{\parallel }\mathrm{.}$

If each subdomain state is reconstructed by solving its interior equations at the supplied trace, define the product backward-Euler energy norm by $\mathrm{\parallel }(z_1,z_2){\mathrm{\parallel }}_{B_1\oplus B_2}^2:=⟨B_1{z}_1,z_1⟩+⟨B_2{z}_2,z_2⟩$ for $z_i\in X_i$. Then

$\mathrm{\parallel }\tilde{u}-u_{\ast }{\mathrm{\parallel }}_{B_1\oplus B_2}\le \sqrt{\mathrm{\parallel }{S}^{-1}\mathrm{\parallel }}\mathrm{\parallel }\rho \mathrm{\parallel }\mathrm{.}$

Let $d=1$ and let $A_t$ be an ALE moving-domain map of ALE moving-domain map and mesh velocity, so $\widehat{\mathrm{\Omega }}$ is a finite union of bounded intervals with pairwise disjoint closures and ${\mathrm{\partial }}_{\xi }{A}_t>0$. For every $C^1$ scalar field $g$ on the moving closure,

$\frac{d}{dt}{\int }_{\mathrm{\Omega }(t)}g(x,t)dx={\int }_{\mathrm{\Omega }(t)}{\mathrm{\partial }}_{t}g(x,t)dx+{\sum }_{x\in \mathrm{\partial }\mathrm{\Omega }(t)}g(x,t)w_m(x,t)n(x),$

where $n=-1$ at each left endpoint and $n=+1$ at each right endpoint. This dimension-one branch is choice-free.

Assume Countable Choice and $d\ge 2$. Let $A_t$ be the ALE moving-domain map of ALE moving-domain map and mesh velocity, and let $g(x,t)$ and its first derivatives be continuous on the moving closure. Then, for each $t\in (0,T)$,

$\frac{d}{dt}{\int }_{\mathrm{\Omega }(t)}g(x,t)dx={\int }_{\mathrm{\Omega }(t)}{\mathrm{\partial }}_{t}g(x,t)dx+{\int }_{\mathrm{\partial }\mathrm{\Omega }(t)}g(x,t)w_m(x,t)\cdot n(x,t)dS\mathrm{.}$

Here $n$ is the outward normal from the transported finite piecewise-$C^1$ boundary presentation. This is a continuous transport identity; no discrete high-dimensional geometric conservation estimate follows from it.

For every input satisfying Conservative upwind ALE finite-volume step, the procedure completes after finitely many operations and returns exactly one of mesh_crossing (with no updated state, if some $J_i^{n+1}\le 0$) or success (with a unique updated array, if all $J_i^{n+1}>0$). Whenever it returns success, it preserves total periodic mass, ${\sum }_i{J}_i^{n+1}{\bar{u}}_i^{n+1}={\sum }_i{J}_i^n{\bar{u}}_i^n$. It also preserves every constant state: if ${\bar{u}}_i^n=U$ for every cell, then ${\bar{u}}_i^{n+1}=U$ for every cell. The mass and constant-state properties require no CFL condition.