Linear scheme stability and Lax equivalence — Examples

Let $D$ be the finite-support sequences. By [F4], $D$ is dense in $X$; it is invariant under $U(t)$ because $U(t)$ multiplies coordinates and does not add support. The heat IVP and its finite-time bound are supplied by [F3].

Since $m=\nu +1$, ${\tau }_m=m^{-2}>0$ and ${\tau }_m\to 0$ as $\nu \to \mathrm{\infty }$. Let $P_m$ be truncation to coordinates $0,\dots ,m-1$ and $R_m$ inclusion. Their norms are at most $1$ by the coordinate supremum estimates and at least $1$ by testing $e_0$, so both have norm $1$ by [F7]. Also $P_m{R}_m=I_{G_m}$ and $\ker P_m$ is invariant under $U({\tau }_m)$ because $U$ is diagonal. The finite maximum $L_m:={\max }_{0\le j<m}\mathrm{\mid }1-{\tau }_m{j}^2\mathrm{\mid }$ bounds every diagonal multiplier, so $\mathrm{\parallel }{Q}_{m}g{\mathrm{\parallel }}_{\mathrm{\infty }}\le L_m\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $Q_m$ is bounded. Thus [F2] defines the common-space step $S_m$.

For $j<m$, $0\le 1-j^2\mathrm{/}{m}^2\le 1$; on coordinates $j\ge m$ the lift uses the multipliers $\exp (-j^2\mathrm{/}{m}^2)\in (0,1]$ from [F3]. Hence every coordinate multiplier of $S_m$ lies in $[0,1]$. Therefore $\mathrm{\parallel }{S}_{m}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ for every $x\in c_0$, and [F7] gives $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$. The bound $\mathrm{\parallel }{S}_m^{n}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ holds at $n=0$; if it holds at $n$, applying the contraction estimate to $S_m^{n}x$ proves it at $n+1$. Induction [F8] gives it for every $n$, and then [F7] gives $\mathrm{\parallel }{S}_m^n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$. The family is stable on every finite time interval.

Fix $v\in D$, supported in coordinates $0,\dots ,N$. For $m>N$, direct evaluation and $0<\exp (-r{\tau }_m{j}^2)\le 1$ give ${\max }_{\begin{array}{c}r\in \mathbb{N}\\ r{\tau }_m\le T\end{array}}\frac{\mathrm{\parallel }(S_m-U({\tau }_m))U(r{\tau }_m)v{\mathrm{\parallel }}_{\mathrm{\infty }}}{{\tau }_m}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_{0\le j\le N}\frac{\mathrm{\mid }1-{\tau }_m{j}^2-\exp (-{\tau }_m{j}^2)\mathrm{\mid }}{{\tau }_m}\mathrm{.}$ For $j=0$ the quotient is zero. For each fixed $1\le j\le N$, put $z={\tau }_m{j}^2$; [F5] gives $\frac{\mathrm{\mid }1-z-\exp (-z)\mathrm{\mid }}{{\tau }_m}=j^2\mid \frac{1-\exp (-z)}{z}-1\mid \to 0$, because the quotient tends to ${\exp }^{\mathrm{\prime }}(0)=1$ as $z\to 0$. The maximum of these finitely many quotients tends to zero: for any tolerance choose a threshold for each $j$ (or use threshold $1$ if there is no positive $j$), then take the largest threshold. Multiplication by the fixed $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ preserves this limit. The bound is independent of $r$, so the family is consistent on $D$.

By [F6] and the explicit DC assumption [A1], this consistent stable family converges to the heat semigroup in the common supremum norm, uniformly over grid times in each bounded time interval.

Let $D$ be the finite-support sequences. It is dense by [F4] and invariant under the diagonal semigroup [F3]. Since $m=\nu +1$, ${\tau }_m=m^{-1}>0$ and ${\tau }_m\to 0$. For each $m$, take $P_m$ to be coordinate truncation and $R_m$ inclusion. The estimate $\mathrm{\parallel }{P}_{m}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ and identity $\mathrm{\parallel }{R}_{m}g{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$, together with testing $e_0$ in [F6], give $\mathrm{\parallel }{P}_m\mathrm{\parallel }=\mathrm{\parallel }{R}_m\mathrm{\parallel }=1$. Then $P_m{R}_m=I$ and $\ker P_m$ is invariant under diagonal $U({\tau }_m)$. The finite maximum $L_m:={\max }_{0\le j<m}\mathrm{\mid }1-{\tau }_m{j}^2\mathrm{\mid }$ gives $\mathrm{\parallel }{Q}_{m}g{\mathrm{\parallel }}_{\mathrm{\infty }}\le L_m\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$, so $Q_m$ is bounded. Thus [F2] gives a bounded common-space step.

Fix $v\in D$ with support in coordinates $0,\dots ,N$. For $m>N$, every supported coordinate is updated by $1-{\tau }_m{j}^2$; since $0<\exp (-r{\tau }_m{j}^2)\le 1$, we have ${\max }_{\begin{array}{c}r\in \mathbb{N}\\ r{\tau }_m\le T\end{array}}\frac{\mathrm{\parallel }(S_m-U({\tau }_m))U(r{\tau }_m)v{\mathrm{\parallel }}_{\mathrm{\infty }}}{{\tau }_m}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_{0\le j\le N}\frac{\mathrm{\mid }1-{\tau }_m{j}^2-\exp (-{\tau }_m{j}^2)\mathrm{\mid }}{{\tau }_m}\mathrm{.}$ For $j=0$ the quotient vanishes. For each fixed $1\le j\le N$, with $z={\tau }_m{j}^2$, the quotient equals $j^2\mathrm{\mid }(1-\exp (-z))\mathrm{/}z-1\mathrm{\mid }$, which tends to zero by [F5]. The finite maximum tends to zero: for any tolerance choose a threshold for each of the finitely many $j$ (or use threshold $1$ if there is no positive $j$), then take the largest threshold. Multiplication by fixed $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ preserves the limit, uniformly in the permitted $r$. Thus the family is consistent.

On the highest grid mode $e_{m-1}$, the one-step multiplier is $1-\frac{(m-1{)}^2}{m}=3-m-\frac{1}{m}\mathrm{.}$ For $m\ge 4$ its magnitude is $m-3+1\mathrm{/}m$, which tends to infinity. Given any $T>0$, all sufficiently large $m$ also satisfy ${\tau }_m=1\mathrm{/}m\le T$; then the allowed power $n=1$ already has $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\ge m-3+1\mathrm{/}m$ by testing the unit vector $e_{m-1}$ in [F6]. No uniform finite-time power bound can hold, so the family is unstable.

Let $D$ be the finite-support sequences. By [F4], $X$ is Banach; the identity maps $U(t)=I_X$ form a strongly continuous semigroup with finite-time bound $M_T=1$. By [F5], $D$ is dense and it is invariant under $U$. The positive steps ${\tau }_m=1\mathrm{/}m$ tend to zero, so this is common-space scheme data.

The positive coordinate weights make the displayed maximum a norm: homogeneity and definiteness hold coordinatewise, and the triangle inequality follows by applying the scalar triangle inequality in every coordinate before taking the maximum. The truncation satisfies $\mathrm{\parallel }{P}_{m}x{\mathrm{\parallel }}_{G_m}\le m\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$, and inclusion satisfies $\mathrm{\parallel }{R}_{m}g{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }g{\mathrm{\parallel }}_{G_m}$, so both are bounded. Also $P_m{R}_m=I_{G_m}$ and $\ker P_m$ is invariant under $U=I$. For $g={\sum }_{j<m}{g}_j{e}_j$, the zeroth output coordinate is $g_0+m{g}_{m-1}$, coordinates $1$ through $m-2$ if present remain $g_j$, and coordinate $m-1$ becomes $0$. Since $\mathrm{\mid }{g}_0\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{G_m}$, $m\mathrm{\mid }{g}_{m-1}\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{G_m}$, and $\mathrm{\mid }{g}_j\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_{G_m}$ for $1\le j\le m-2$, it follows that $\mathrm{\parallel }{Q}_{m}g{\mathrm{\parallel }}_{G_m}\le 2\mathrm{\parallel }g{\mathrm{\parallel }}_{G_m}$, including $m=2$. Thus $Q_m$ is bounded and the grid lift [F2] is defined. Also $Q_m^2=Q_m$: it fixes each $e_j$ for $j<m-1$, and $Q_m^2{e}_{m-1}=Q_m(m{e}_0)=m{e}_0=Q_m{e}_{m-1}$. By induction [F7], $Q_m^n=Q_m$ for every $n\ge 1$: the base $n=1$ is immediate, and $Q_m^n=Q_m$ implies $Q_m^{n+1}=Q_m{Q}_m=Q_m^2=Q_m$. Meanwhile $Q_m^0=I$. By [F6], $\mathrm{\parallel }{Q}_m^n{\mathrm{\parallel }}_{G_m\to G_m}\le 2$ for every $n\ge 0$, so the grid updates are stable in their mesh norms.

Fix $v\in D$ and choose $N\in \mathbb{N}$ such that $v_j=0$ for every $j>N$ (take $N=0$ for the zero sequence). For $m\ge N+2$, one has $P_{m}v=v$, the high coordinate $v_{m-1}=0$ since $m-1>N$, and $Q_{m}v=v$. Since $U=I$, the lift gives $S_{m}v=v$. Thus the local defect is zero on every $v\in D$ for all sufficiently large $m$, which proves consistency.

For $e_{m-1}$, the lift gives $S_m{e}_{m-1}=m{e}_0$. Its common-space norm is therefore $m$, while $\mathrm{\parallel }{e}_{m-1}{\mathrm{\parallel }}_{\mathrm{\infty }}=1$. Testing this unit vector in [F6] gives $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_{X\to X}\ge m$. For every $T>0$, all sufficiently large $m$ have ${\tau }_m\le T$, so the common-space family is not stable.

The same vector satisfies $\mathrm{\parallel }{e}_{m-1}{\mathrm{\parallel }}_{G_m}=m$ but $\mathrm{\parallel }{R}_m{e}_{m-1}{\mathrm{\parallel }}_{\mathrm{\infty }}=1$; thus no positive uniform lower comparison constant $a$ exists. Also $\mathrm{\parallel }{P}_m{\mathrm{\parallel }}_{X\to G_m}\ge m$ by testing the unit vector $e_{m-1}$ in [F6]. The hypotheses needed in [F3] fail, exactly where the grid norm ceases to control the reconstructed common-space norm and the restriction norm. The grid stability calculation therefore cannot certify common-space stability.