TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Local truncation residual for the θ heat scheme

Statement

Let $Ω = (0, L_{x}) \times (0, L_{y})$ , $T > 0$ , $N_{x}, N_{y} \geq 2$ , $M \geq 1$ , $h_{x} = L_{x} / N_{x}$ , $h_{y} = L_{y} / N_{y}$ , and $Δ t = T / M$ . Let $κ_{x}, κ_{y} > 0$ , $θ \in [0, 1]$ , and suppose $u$ satisfies

u_t-\kappa_xu_{xx}-\kappa_yu_{yy}=f$$ on $\overline\Omega\times[0,T]$, with exact sampled initial and Dirichlet rows in the scheme of [Fully discrete θ heat finite-difference scheme](/item/def-parabolic-theta-finite-difference-scheme). Assume $u_{tt},u_{ttt},u_{xxxx},u_{yyyy}$ are continuous on the closed cylinder, and write $M_{tt},M_{ttt},M_{x4},M_{y4}$ for their respective sup norms there. If $r^n$ is the exact grid sample at time $t_n$, the complete interior residual is bounded at every time step by

|\tau^n|\infty\le \Delta t M{tt}+\frac{\kappa_xh_x^2M_{x4}+\kappa_yh_y^2M_{y4}}{12}\quad(\theta\ne\tfrac12),

a n d

|\tau^n|\infty\le\frac{(\Delta t)^2M{ttt}}{12}+\frac{\kappa_xh_x^2M_{x4}+\kappa_yh_y^2M_{y4}}{12}\quad(\theta=\tfrac12),

where $\tau^n$ is defined by substituting the exact samples and the exact boundary rows into the fully discrete equation, including its interior rows adjacent to the boundary.

Facts & Assumptions

Given: The rectangular domain and finite mesh above; $κ_{x}, κ_{y} > 0$ ; $θ \in [0, 1]$ ; an exact solution with the stated continuous derivatives satisfying $u_{t} - κ_{x} u_{x x} - κ_{y} u_{y y} = f$ ; and exact initial and boundary samples.

[F1]

Fully discrete θ heat finite-difference scheme defines the centered heat stencil, the endpoint-weighted source $(1 - θ) f^{n} + θ f^{n + 1}$ , and boundary contributions from both endpoint rows.

Proof

1.1F1givenalgebra

At each interior node set $r_{i j}^{n} = u (x_{i}, y_{j}, t_{n})$ and define $τ^{n} = (r^{n + 1} - r^{n}) / Δ t + A_{h}^{B_{θ}} ((1 - θ) r^{n} + θ r^{n + 1}) - ((1 - θ) f^{n} + θ f^{n + 1})$ , where $B_{θ} = (1 - θ) B^{n} + θ B^{n + 1}$ and $A_{h}^{B_{θ}}$ applies the centered stencil using that exact boundary row. Applying the PDE at $t_{n}$ and $t_{n + 1}$ rewrites this as the weighted average of the two spatial stencil defects plus $(r^{n + 1} - r^{n}) / Δ t - ((1 - θ) u_{t}^{n} + θ u_{t}^{n + 1})$ .

2.1F1givenstep 1.1algebra

For a one-variable $C^{4}$ function $v$ , Taylor expansion at an interior node gives $(v (x + h) - 2 v (x) + v (x - h)) / h^{2} - v^{''} (x) = h^{2} (v^{(4)} (ξ_{+}) + v^{(4)} (ξ_{-})) / 24$ for some $ξ_{+} \in (x, x + h)$ and $ξ_{-} \in (x - h, x)$ . Thus the absolute error is at most $h^{2} ∥ v^{(4)} ∥_{\infty} / 12$ . The stencil uses the exact boundary samples at the first and last interior rows, so the same Taylor estimate applies there. Applying it in $x$ and $y$ bounds each spatial defect in step 1.1 by $(κ_{x} h_{x}^{2} M_{x 4} + κ_{y} h_{y}^{2} M_{y 4}) / 12$ .

2.2givenstep 1.1algebra

Put $g (t) = u_{t} (x_{i}, y_{j}, t)$ . The quotient $(u (t_{n + 1}) - u (t_{n})) / Δ t$ is the average of $g$ on the time interval. For $θ \neq 1 / 2$ , both this average and $(1 - θ) g (t_{n}) + θ g (t_{n + 1})$ lie between the minimum and maximum of $g$ on that interval; their difference is therefore at most its oscillation, which is at most $Δ t M_{t t}$ . For $θ = 1 / 2$ , let $L$ be the linear interpolant of $g$ at the two endpoints. The interpolation remainder satisfies $∣ g (t) - L (t) ∣ \leq M_{t t t} (t - t_{n}) (t_{n + 1} - t) / 2$ . Integrating and dividing by $Δ t$ gives $∣ (Δ t)^{- 1} \int_{t_{n}}^{t_{n + 1}} g - (g (t_{n}) + g (t_{n + 1})) / 2 ∣ \leq (Δ t)^{2} M_{t t t} / 12$ . These are the required bounds for the temporal term in step 1.1.

3.1step 1.1step 2.1step 2.2algebra∎

Combining the two spatial bounds from step 2.1 with the temporal bound from step 2.2 in the residual identity of step 1.1 gives the two asserted estimates at every interior node. Taking the maximum over nodes and time steps proves the stated complete-residual bounds.

Remarks

The derivatives are assumed up to the closed space-time boundary, so the proof includes boundary-adjacent interior rows. Exact boundary samples are essential; no result for a lower-order or different boundary closure is asserted. No choice axiom is used.

Depends on

Fully discrete θ heat finite-difference scheme

Used by

Global weighted L2 error from the complete parabolic residual Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Endre Süli, B6.1 Numerical Solution of Partial Differential Equations: A brief introduction to the theory of finite difference approximation of PDEs (standard reference, not scraped)
Benjamin Seibold, MIT 18.336, Lecture 14: Von Neumann Stability Analysis (Spring 2009) (standard reference, not scraped)
Martin Krosche, Hermann G. Matthies, and Oliver Kayser-Herold, Lecture Notes: Introduction to PDEs and Numerical Methods, Technical University Braunschweig, Revision 1.1 (2008), hosted by University of Jyväskylä (standard reference, not scraped)