Common Banach-space interface for a linear IVP and its time schemes

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.