Fixed-matrix global orders and certified-action error propagation

Given: The matrix, grid, solution, tube, regularity, Lipschitz and semigroup bounds, and numerical action certificates in the Statement.

First-order systems, initial value problems, and solutions on intervals: the exact solution is differentiable and satisfies $y^{\mathrm{\prime }}(t)=Ay(t)+g(t,y(t))$.

Exponential Euler has a second-order local defect: the exact-start exponential-Euler defect is at most $C_E{h}^2$.

The two-stage ETD2 method has a third-order local defect: the exact-start ETD2 defect is at most $C_2{h}^3$.

One-step stability after removing the exact linear flow: after the change $z(t)=e^{-tA}y(t)$, the exponential-Euler and ETD2 one-step maps have stability constants $L_E$ and $L_2$ in the form $1+L_{p}h$.

Certified action bounds control one exponential-integrator step: when the action algorithm is certified, $\mathrm{\parallel }{\tilde{\mathrm{\Psi }}}_{h,n}(v)-{\mathrm{\Psi }}_{h,n}(v){\mathrm{\parallel }}_2\le {\delta }_n$.

A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Every finite-dimensional normed space is Banach, The exponential series of a bounded operator, and Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: on ${\mathbb{C}}^d$ the matrix exponential is continuous and differentiable, with $(e^{tA}{)}^{\mathrm{\prime }}=A{e}^{tA}=e^{tA}A$ and the semigroup law, and finite-dimensional products obey the product rule.

The real exponential function and the number $e$ by a power series and $1+x\le \exp (x)$ for every real $x$, hence $(1-p{)}^m\le \exp (-mp)$: for $x\ge 0$, $1+x\le e^x$.

The exponential addition formula $\exp (x+y)=\exp (x)\exp (y)$ and The exponential function is strictly increasing: $e^{x+y}=e^x{e}^y$ and $e^x\le e^y$ whenever $x\le y$.

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, and The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||^2: the vector norm is Euclidean, the matrix norm is induced, and $\mathrm{\parallel }Bv{\mathrm{\parallel }}_2\le \mathrm{\parallel }B{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$.

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation: the realified time-state space ${\mathbb{R}}^{1+2d}$ with its standard Euclidean norm is a normed space.

A compact set and a disjoint closed set have a positive norm-distance gap: a compact subset of an open subset of a finite-dimensional normed space has positive distance from its closed complement when that complement is nonempty; if it is empty, every positive radius is a margin.

Metric continuity characterisations, with countable choice for the sequential converse: a continuous map between metric spaces has open preimages of open sets.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, the componentwise map ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear, with $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. It preserves real-time difference quotients and their limits, so the real one-step theorem applies to the realified system and its transformed state without changing the error norm.

The exponential series of a bounded operator gives $\mathrm{\parallel }{e}^{-sA}{\mathrm{\parallel }}_2\le e^{s\mathrm{\parallel }A{\mathrm{\parallel }}_2}$ for $0\le s\le T$. Since $s\mathrm{\parallel }A{\mathrm{\parallel }}_2\le T\mathrm{\parallel }A{\mathrm{\parallel }}_2$, scalar monotonicity (The exponential function is strictly increasing) gives $e^{s\mathrm{\parallel }A{\mathrm{\parallel }}_2}\le e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. Thus the set defining $K_{-}$ contains $\mathrm{\parallel }{I}_d{\mathrm{\parallel }}_2$ at $s=0$ and has that finite upper bound; Dedekind completeness: the least-upper-bound property supplies its finite supremum. Likewise $\omega s\le \max (\omega ,0)T$, so the assumed semigroup bound and the same scalar monotonicity give $K_{+}\le M{e}^{\max (\omega ,0)T}$ and a finite supremum for its nonempty defining set.

Global error from uniform one-step defect and stability: on a common compact tube, exact-start local defect $C{h}^{p+1}$, one-step stability $1+Lh$, and exact initial values imply the fixed-time global bound $C\mathrm{\Gamma }(L,T)h^p$.

The transformed open domain and compact tube satisfy the hypotheses stated above; the realified norm and coordinate calculus are as in [F13].