Linear scheme stability and Lax equivalence

Fix a scalar field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. A common-space linear IVP interface consists of a Banach space $X$ over $\mathbb{K}$ and a strongly continuous semigroup $U(t)\in \mathcal{B}(X)$, $t\ge 0$, as defined in Strongly continuous semigroup, together with a finite-time bound: for each $T>0$ there is $M_T<\mathrm{\infty }$ such that ${\sup }_{0\le t\le T}\mathrm{\parallel }U(t)\mathrm{\parallel }\le M_T\mathrm{.}$ The solution with initial value $x\in X$ is $u(t)=U(t)x$. This definition records the solution operators and their continuous dependence in the same norm; it makes no claim about a generator domain.

A consistency subspace is a dense linear subspace $D\subseteq X$ satisfying $U(t)D\subseteq D$ for every $t\ge 0$. Here dense means ${\bar{D}}^X=X$, and invariant means the displayed inclusion. It need not be the domain of a differential generator.

A common-space time-scheme family is a sequence of positive time steps $({\tau }_{\nu }{)}_{\nu \in \mathbb{N}}$ with ${\tau }_{\nu }\to 0$ and bounded linear maps $S_{\nu }:X\to X$ in the sense of A bounded linear operator between normed spaces. Write $S_{\nu }^0=I_X$ and recursively set $S_{\nu }^{n+1}=S_{\nu }{S}_{\nu }^n$ for $n\in \mathbb{N}$. These powers are uniquely defined by the recursion theorem (The recursion theorem) on the set $\mathcal{B}(X)$ of bounded linear operators (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators): for fixed $S_{\nu }$, the map $T\mapsto S_{\nu }T$ preserves $\mathcal{B}(X)$ because composition preserves linearity and multiplies operator bounds, and $I_X\in \mathcal{B}(X)$ with bound $1$. All scheme and solution errors below use the norm of this same space $X$; write $\mathrm{\parallel }\cdot \mathrm{\parallel }$ for that norm and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ for the operator norm of The operator norm as the least bound and as the unit-sphere or unit-ball supremum.

The family is consistent on $D$ when for every $x\in D$ and $T>0$, ${\lim }_{\nu \to \mathrm{\infty }}{\max }_{\begin{array}{c}r\in \mathbb{N}\\ r{\tau }_{\nu }\le T\end{array}}\frac{\mathrm{\parallel }(S_{\nu }-U({\tau }_{\nu }))U(r{\tau }_{\nu })x\mathrm{\parallel }}{{\tau }_{\nu }}=0.$ The maximum is over a nonempty finite set because it includes $r=0$. This is the local defect divided by one time step, uniformly over the exact orbit on the finite time interval.

The family is stable on finite time intervals when for every $T>0$ there is $C_T<\mathrm{\infty }$ such that $\mathrm{\parallel }{S}_{\nu }^n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C_T\text{whenever }\nu ,n\in \mathbb{N}\text{ and }n{\tau }_{\nu }\le T\mathrm{.}$ It is convergent on finite time intervals when for every $x\in X$ and $T>0$, ${\lim }_{\nu \to \mathrm{\infty }}{\max }_{\begin{array}{c}n\in \mathbb{N}\\ n{\tau }_{\nu }\le T\end{array}}\mathrm{\parallel }{S}_{\nu }^{n}x-U(n{\tau }_{\nu })x\mathrm{\parallel }=0.$ The maximum convention includes time zero. This locally uniform-in-grid-time convergence is the hypothesis used for the reverse implication in the equivalence theorem below.

• A finite-dimensional grid update can be used in this interface only after it has been represented by bounded operators on the common space. The exact projection/reconstruction lift and the norm bounds needed to transfer stability are stated separately in Finite-grid restriction, reconstruction, and common-space lift and Grid and common-space stability agree under uniform norm comparison.
• A spectral-radius, von Neumann, or CFL calculation is not the stability condition in this definition; stability is the displayed uniform bound on powers in the declared common norm.

Let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ and let $X=c_0(\mathbb{K})$ with the supremum norm, as in The sequence spaces c_0 and ell-infinity. For $t\ge 0$ define $U(t)x=(\exp (-j^{2}t)x_j{)}_{j\in \mathbb{N}}\mathrm{.}$ Then $U(t)$ is a strongly continuous semigroup on the Banach space $X$ and $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1(t\ge 0)\mathrm{.}$

Let $(X,U,D,({\tau }_{\nu }))$ be a common-space IVP interface from Common Banach-space interface for a linear IVP and its time schemes. A finite grid space $G_{\nu }$ is a normed vector space spanned by a possibly empty finite list over the same field $\mathbb{K}$. A grid realization consists, for each $\nu$, of bounded linear maps $P_{\nu }:X\to G_{\nu },R_{\nu }:G_{\nu }\to X,Q_{\nu }:G_{\nu }\to G_{\nu },$ where bounded linearity is in the sense of A bounded linear operator between normed spaces, such that $P_{\nu }{R}_{\nu }=I_{G_{\nu }}$ and $U({\tau }_{\nu })(\ker P_{\nu })\subseteq \ker P_{\nu },\ker P_{\nu }:=\{x\in X:P_{\nu }x=0\}\mathrm{.}$ Here $P_{\nu }$ is restriction to grid data, $R_{\nu }$ reconstructs grid data in $X$, and $Q_{\nu }$ is the one-step grid update.

The common-space lift of the grid update is $S_{\nu }:=R_{\nu }{Q}_{\nu }{P}_{\nu }+U({\tau }_{\nu })(I_X-R_{\nu }{P}_{\nu })\mathrm{.}$ The first summand advances the reconstructed grid component; the second continues the exact semigroup on the complementary component. The lift is bounded because all factors are bounded. More explicitly, if $c_P,c_R,c_Q$ bound $P_{\nu },R_{\nu },Q_{\nu }$ and $c_U$ bounds $U({\tau }_{\nu })$, then $\mathrm{\parallel }{S}_{\nu }x\mathrm{\parallel }\le (c_R{c}_Q{c}_P+c_U(1+c_R{c}_P))\mathrm{\parallel }x\mathrm{\parallel }$ by the factor bounds and the triangle inequality. This finite bound proves that $S_{\nu }$ is bounded. The invariance hypothesis on $\ker P_{\nu }$ is used later to identify the lift's powers when comparing grid and common-space stability. This definition is an analysis interface; it does not assert that evaluating the complementary semigroup is part of a finite-grid implementation.

Assume the Axiom of Dependent Choice ($\mathrm{D}\mathrm{C}$). Let a common-space IVP interface and bounded time-scheme family be given as in Common Banach-space interface for a linear IVP and its time schemes. If the scheme is consistent on its dense invariant subspace, then it is convergent on every finite time interval if and only if it is stable there, with stability, consistency, and convergence all measured in the same norm of $X$.

Let a finite-grid lift be as in Finite-grid restriction, reconstruction, and common-space lift over the common Banach-space interface of Common Banach-space interface for a linear IVP and its time schemes. Assume there are constants $a,b,P>0$, independent of $\nu$, such that for every $g\in G_{\nu }$ and $x\in X$, $a\mathrm{\parallel }g{\mathrm{\parallel }}_{G_{\nu }}\le \mathrm{\parallel }{R}_{\nu }g{\mathrm{\parallel }}_X\le b\mathrm{\parallel }g{\mathrm{\parallel }}_{G_{\nu }},\mathrm{\parallel }{P}_{\nu }{\mathrm{\parallel }}_{X\to G_{\nu }}\le P\mathrm{.}$ Then the lifted family $(S_{\nu })$ is stable on finite time intervals if and only if the grid-update family $(Q_{\nu })$ is stable in the declared grid norms on finite time intervals.

Here grid stability means that for every $T>0$ there is $C_T<\mathrm{\infty }$ such that $\mathrm{\parallel }{Q}_{\nu }^n{\mathrm{\parallel }}_{G_{\nu }\to G_{\nu }}\le C_T$ whenever $\nu ,n\in \mathbb{N}$ and $n{\tau }_{\nu }\le T$.