Finite-time continuum limit of the cyclic spring chain: uniform $O(a^{3\mathrm{/}2})$ nodal and uniform interpolation convergence

Let $L>0$, $c>0$, let $N\ge 2$ be an integer and $a:=L\mathrm{/}N$, and let $[0,T]$ be a compact time interval, with $T\ge 0$. Let

$u:[0,T]\times \mathbb{R}\to \mathbb{R}$

be $C^2$ in $t$ and $C^4$ in $x$ ($C^k$ Euclidean maps and diffeomorphisms), $L$-periodic in $x$, and satisfy $u_{tt}=c^2{u}_{xx}$ on $[0,T]\times \mathbb{R}$; put

$M:={\sup }_{[0,T]\times \mathbb{R}}\mathrm{\mid }{\mathrm{\partial }}_x^{4}u\mathrm{\mid }<+\mathrm{\infty },M_2:={\sup }_{[0,T]\times \mathbb{R}}\mathrm{\mid }{\mathrm{\partial }}_x^{2}u\mathrm{\mid }<+\mathrm{\infty }\mathrm{.}$

Let $q:[0,T]\to {\mathbb{R}}^N$ be the (by (i), unique) solution of the cyclic spring-chain system of The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class with the sampled initial data $q_j(0)=u(0,ja)$, $q_j^{\mathrm{\prime }}(0)=u_t(0,ja)$, and put

$e_j(t):=q_j(t)-u(t,ja),C:=\frac{c^2\sqrt{L}M}{12}\mathrm{.}$

Then:

Here smoothness on the closed time interval means restriction of a map defined on an open neighbourhood, with the indicated derivatives extending continuously to the interval. A chain solution on $[0,T]$ is the restriction of a solution on an open time interval containing $[0,T]$; the sampled derivative $q_j\mathrm{\prime }(0)$ is the derivative of that solution. Thus derivatives remain defined when $T=0$. Uniqueness concerns the restricted state $(q,q\mathrm{\prime })$.

(i) the solution $q$ exists and is unique on $[0,T]$;

(ii) for every $t\in [0,T]$,

$\mathrm{\parallel }e(t){\mathrm{\parallel }}_a\le \frac{C{a}^2{t}^2}{2},\mathrm{\parallel }{e}^{\mathrm{\prime }}(t){\mathrm{\parallel }}_a\le C{a}^{2}t;$

(iii) consequently ${\max }_j\mathrm{\mid }{e}_j(t)\mathrm{\mid }\le \frac{1}{2}C{a}^{3\mathrm{/}2}{t}^2$ for $t\in [0,T]$, so the nodal error tends to $0$ uniformly as $a\to 0$;

(iv) let $I_{a}q(t,\cdot )$ be the piecewise-linear function on $[0,L]$ that equals $q_j(t)$ at $x=ja$ and is extended $L$-periodically; then for the periodic extension of $u$,

${\sup }_{x\in [0,L]}\mid I_{a}q(t,x)-u(t,x)\mid \le {\max }_j\mathrm{\mid }{e}_j(t)\mathrm{\mid }+\frac{M_2{a}^2}{8}(0\le t\le T),$

so the interpolated chain displacement converges to $u$ uniformly on $[0,T]\times [0,L]$ as $a\to 0$.

• The convergence is finite-time and uniform only for the stated smooth $L$-periodic regime; the rate is the nodal $O(a^{3\mathrm{/}2})$ together with the interpolation error $O(a^2)$, and no uniform bound on $\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$ is claimed beyond what the estimates give. Rough data, non-periodic boundary conditions and nonlinear chains are separate problems.

• The mechanical interpretation $m=\rho a$, $k=T_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}\mathrm{/}a$, $c^2=k{a}^2\mathrm{/}m$ is a model reading of the parameters; the theorem compares the two mathematical evolutions and asserts no physical identification of the chain with any material string.

• The estimate is derived from the energy identity, so it requires only the perturbation $R$ to be continuous and the chain to solve exactly the linear system; no numerical integrator error is analyzed, and floating-point roundoff is not part of the statement.