A boundary-aware discrete maximum principle for the θ scheme

Put $s={\mu }_x+{\mu }_y$ and let $V^k$ denote the full grid values, equal to $U^k$ in the interior and $B^k$ on the boundary. At each interior node the zero-source update is $(1+2\theta s)U_{ij}^{n+1}-\theta {\mu }_x(V_{i+1,j}^{n+1}+V_{i-1,j}^{n+1})-\theta {\mu }_y(V_{i,j+1}^{n+1}+V_{i,j-1}^{n+1})=[1-2(1-\theta )s]U_{ij}^n+(1-\theta ){\mu }_x(V_{i+1,j}^n+V_{i-1,j}^n)+(1-\theta ){\mu }_y(V_{i,j+1}^n+V_{i,j-1}^n)$. The right side is known; its center coefficient is nonnegative by the stated condition, and its neighbor coefficients are nonnegative because $\theta \in [0,1]$ and ${\mu }_x,{\mu }_y>0$. Thus it is nonnegative whenever the old interior and boundary values are nonnegative.

Let $W$ be the unique interior solution of the implicit system with nonnegative right-hand side and nonnegative future boundary data, whose existence is [F2]. If an interior value attained a negative minimum $W_{ij}<0$, then every adjacent interior or boundary value would be at least $W_{ij}$, so $W_{ij}+\theta {\mu }_x(2W_{ij}-W_{i+1,j}-W_{i-1,j})+\theta {\mu }_y(2W_{ij}-W_{i,j+1}-W_{i,j-1})\le W_{ij}<0$. This is the corresponding component of the nonnegative right-hand side, a contradiction. Hence $W\ge 0$; for $\theta =0$ the same inequality reduces to $W_{ij}<0$ and gives the contradiction directly.

The zero-source scheme is linear. Let $M_0$ and $m_0$ be respectively the maximum and minimum of the finite collection of initial and prescribed boundary values at all time levels. The constant grid with interior and boundary values $M_0$ solves the zero-source scheme with constant boundary rows $M_0$, since every centered second difference of a constant is zero. Subtracting the numerical solution $U$ gives a solution with initial values $M_0-U^0\ge 0$ and boundary rows $M_0-B^n\ge 0$. Induction using steps 1.1–1.2 gives $U\le M_0$. Likewise, subtracting the constant grid with interior and boundary values $m_0$ from $U$ gives initial values $U^0-m_0\ge 0$ and boundary rows $B^n-m_0\ge 0$, so induction gives $U\ge m_0$. Thus $m_0\le U_{ij}^n\le M_0$ at every interior node and time level; the prescribed boundary nodes satisfy the same bounds by definition.

In the homogeneous-boundary case at $\theta =1\mathrm{/}2$, the stated sufficient maximum-principle restriction reduces to $\mathrm{\Delta }t({\kappa }_x\mathrm{/}{h}_x^2+{\kappa }_y\mathrm{/}{h}_y^2)\le 1$. By [F3] weighted $L^2$ stability holds for every $\mathrm{\Delta }t>0$ under the same zero-boundary setup. Thus the maximum-principle condition is a stricter sufficient condition; this comparison makes no claim that the maximum principle fails outside it.