Second-order global bound under symmetric coupling and a cubic defect

Let $T>0$, $d\ge 1$, let $F:\mathrm{\Omega }\to {\mathbb{R}}^d$ be continuous on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$, and let $Y:[0,T]\to \mathrm{\Omega }$ be an exact solution of the autonomous IVP $Y^{\mathrm{\prime }}=F(Y)$; here exact means that $Y$ is differentiable and satisfies this equation throughout $[0,T]$. Put $H=T\mathrm{/}N$ with $N\ge 1$. Let ${\mathrm{\Psi }}_H$ be a macro map that produces one output from its current input by composing exact subsystem solution maps over subintervals with a coupling interpolant $I_H$; an exact subsystem solution map sends an initial substate to its exact value at the end of that subinterval. Define numerical states by $Y_{n+1}={\mathrm{\Psi }}_H(Y_n)$. Here “symmetric interpolant” means $I_H(1-\theta ;v,u)=I_H(\theta ;u,v)$ for every $\theta \in [0,1]$ and endpoint pair $u,v$; “symmetric step” additionally means ${\mathrm{\Psi }}_{-H}={\mathrm{\Psi }}_H^{-1}$ wherever both sides are defined. Assume the interpolant and macro step are symmetric in these senses, and there is a compact set $\mathcal{K}\subseteq \mathrm{\Omega }$ containing the exact path, all numerical states, and every method stage. Suppose $\mathrm{\parallel }{Y}_0-Y(0){\mathrm{\parallel }}_2\le C_0{H}^2$ and constants $C,L\ge 0$ independent of $N$ satisfy $\mathrm{\parallel }{\mathrm{\Psi }}_H(Y(t_n))-Y(t_{n+1}){\mathrm{\parallel }}_2\le C{H}^3,\mathrm{\parallel }{\mathrm{\Psi }}_H(u)-{\mathrm{\Psi }}_H(v){\mathrm{\parallel }}_2\le (1+LH)\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$ for every $0\le n<N$ and all admitted inputs $u,v\in \mathcal{K}$. Then the global error is at most ${\max }_{0\le n\le N}\mathrm{\parallel }{Y}_n-Y(t_n){\mathrm{\parallel }}_2\le (C_0\exp (LT)+C\frac{\exp (LT)-1}{L})H^2$ when $L>0$, and at most $(C_0+CT)H^2$ when $L=0$. The symmetry assumptions alone do not imply the cubic defect premise.

The proof uses the one-step cubic defect and stability bounds. A specific Strang or other exact-flow multirate construction must establish those bounds for its actual coupling and interpolation before this corollary applies; symmetry by itself is not a local-error estimate.