Fully discrete θ heat finite-difference scheme

Fix a dimension $d\ge 1$, side lengths $L_j>0$, integers $N_j\ge 2$, directional diffusivities ${\kappa }_j>0$, and spacings $h_j=L_j\mathrm{/}{N}_j$. For a finite periodic rectangular grid let

$$G_h^{\mathrm{p}\mathrm{e}\mathrm{r}}=\prod _{j=1}^d(\mathbb{Z}\mathrm{/}{N}_j\mathbb{Z}),⟨v,w{⟩}_h=\left(\prod _{j=1}^d{h}_j\right)\sum _{i\in G_h^{\mathrm{p}\mathrm{e}\mathrm{r}}}{v}_i{w}_i,$$

and define the positive centered heat operator, with indices reduced modulo $N_j$, by $(A_h^{\mathrm{p}\mathrm{e}\mathrm{r}}v{)}_i={\sum }_{j=1}^d{\kappa }_j\frac{2v_i-v_{i+e_j}-v_{i-e_j}}{h_j^2}\mathrm{.}$

For a two-dimensional Dirichlet rectangle, use the nodes and weighted interior inner product of Rectangular grid and Dirichlet flux stencil. At an interior node $(i,j)$ define $(A_h^{B}V{)}_{ij}={\kappa }_x\frac{2V_{ij}-V_{i+1,j}-V_{i-1,j}}{h_x^2}+{\kappa }_y\frac{2V_{ij}-V_{i,j+1}-V_{i,j-1}}{h_y^2},$ where values on boundary neighbours are the prescribed Dirichlet row $B$. Write $A_h^0$ for the square matrix on interior values when the boundary is zero. For any boundary vector $B$, let $c_h(B)$ be the result of applying this stencil to the grid vector that equals $B$ on the boundary and $0$ at interior nodes. Then $A_h^{B}V=A_h^{0}V+c_h(B)$ on interior nodes.

Let $T>0$, an integer $M\ge 1$, $\mathrm{\Delta }t=T\mathrm{/}M$, $t_n=n\mathrm{\Delta }t$, and $\theta \in [0,1]$. Given interior initial data $U^0$, interior source samples $F^n$, and, in the Dirichlet case, boundary rows $B^n$ for $0\le n\le M$ whose time-zero row agrees with the initial grid data, the periodic update is

$$\frac{U^{n+1}-U^n}{\mathrm{\Delta }t}+A_h^{\mathrm{p}\mathrm{e}\mathrm{r}}((1-\theta )U^n+\theta U^{n+1})=(1-\theta )F^n+\theta F^{n+1},0\le n<M\mathrm{.}$$

The Dirichlet update, with the boundary values imposed at every time level, is

$$\frac{U^{n+1}-U^n}{\mathrm{\Delta }t}+A_h^0((1-\theta )U^n+\theta U^{n+1})=(1-\theta )(F^n-c_h(B^n))+\theta (F^{n+1}-c_h(B^{n+1})),0\le n<M\mathrm{.}$$

Thus the interior rows use the exact boundary contributions at both endpoints of the time step. The choices $\theta =0,1\mathrm{/}2,1$ are forward Euler, Crank–Nicolson, and backward Euler applied to the spatially semidiscrete heat equation; this definition does not redefine those ODE time-stepping procedures.

For a homogeneous-source update with periodic rows or zero Dirichlet boundary values, let $Q_h$ be its one-step linear map when that step is well defined, and give it the operator norm induced by $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_h$. The family is grid-norm power stable on $[0,T]$ if there is $C_T<\mathrm{\infty }$, independent of the grid and time step, such that $\mathrm{\parallel }{Q}_h^n{\mathrm{\parallel }}_{h\to h}\le C_T$ whenever $n\mathrm{\Delta }t\le T$. This is a statement in the mesh-dependent finite grid norms defined above.