Invariant range, TVD and time increments for a monotone finite-volume update

For a finite set $J\subset \mathbb{Z}$, the cell-average formula gives ${\sum }_{j\in J}\mathrm{\mid }{U}_{j+1}^0-U_j^0\mathrm{\mid }\le \frac{1}{h}{\int }_{{\cup }_{j\in J}{I}_j}\mathrm{\mid }{u}_0(x+h)-u_0(x)\mathrm{\mid }dx\le V\mathrm{.}$ Taking the supremum over finite $J$ proves the initial discrete variation bound. Every initial cell average lies in $I$, since $a_{-}\le u_0\le a_{+}$ almost everywhere on its cell.

If $a$ or $c$ increases, monotonicity of $F$ in its first or second coordinate makes $H$ nondecreasing in that variable. If $b_2\ge b_1$, coordinate Lipschitz bounds give $H(a,b_2,c)-H(a,b_1,c)\ge (b_2-b_1)(1-\lambda (L_1+L_2))\ge 0.$ Thus $H$ is nondecreasing in all coordinates. Consistency gives $H(s,s,s)=s$.

Suppose $U^n$ takes values in $I$. Coordinatewise comparison with the constant triples $(a_{-},a_{-},a_{-})$ and $(a_{+},a_{+},a_{+})$, followed by step 1.2, yields $a_{-}=H(a_{-},a_{-},a_{-})\le U_j^{n+1}\le H(a_{+},a_{+},a_{+})=a_{+}$. Induction from step 1.1 proves the range bound at every time.

Let $S$ denote the sequence update. If $x,y\in I^{\mathbb{Z}}$ and ${\sum }_j\mathrm{\mid }{x}_j-y_j\mathrm{\mid }<\mathrm{\infty }$, monotonicity gives $\mathrm{\mid }Sx-Sy\mathrm{\mid }\le S(x\vee y)-S(x\wedge y)$ coordinatewise. The difference on the right is summable: the update formula and the coordinate Lipschitz bounds make its flux differences summable. For any $z,w$ with ${\sum }_j\mathrm{\mid }{z}_j-w_j\mathrm{\mid }<\mathrm{\infty }$, summing the update over $-N\le j\le N$ yields ${\sum }_{j=-N}^N(S{z}_j-S{w}_j)={\sum }_{j=-N}^N(z_j-w_j)-\lambda (d_N-d_{-N-1}),d_j:=F(z_j,z_{j+1})-F(w_j,w_{j+1})\mathrm{.}$ The coordinate Lipschitz bounds make ${\sum }_j\mathrm{\mid }{d}_j\mathrm{\mid }<\mathrm{\infty }$, so both boundary terms tend to zero. The update difference is absolutely summable, and hence ${\sum }_j(S{z}_j-S{w}_j)={\sum }_j(z_j-w_j)$. Applying this to $z=x\vee y$ and $w=x\wedge y$ proves $\mathrm{\parallel }Sx-Sy{\mathrm{\parallel }}_1\le \mathrm{\parallel }x-y{\mathrm{\parallel }}_1$.

The initial neighbor difference is summable by step 1.1. The update commutes with integer shifts because the same $H$ is applied at every index. Apply step 2.2 to $x=U^n$ and its shift $y_j=U_{j+1}^n$ to obtain ${\sum }_j\mathrm{\mid }{U}_{j+1}^{n+1}-U_j^{n+1}\mathrm{\mid }\le {\sum }_j\mathrm{\mid }{U}_{j+1}^n-U_j^n\mathrm{\mid }\mathrm{.}$ Induction proves the stated TVD bound.

Adding and subtracting $F(U_j^n,U_j^n)$ in the update and using the two coordinate Lipschitz bounds gives $\mathrm{\mid }{U}_j^{n+1}-U_j^n\mathrm{\mid }\le \lambda (L_2\mathrm{\mid }{U}_{j+1}^n-U_j^n\mathrm{\mid }+L_1\mathrm{\mid }{U}_j^n-U_{j-1}^n\mathrm{\mid })\mathrm{.}$ Multiply by $h\mathrm{/}\tau =1\mathrm{/}\lambda$, sum over $j$, and use step 3.1 to obtain the time-increment estimate.