TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Second-order local residual of the periodic wave benchmark

Physical scope

A consistency estimate for a smooth nondimensionalized scalar wave on the periodic circle. It is not a global error or Einstein-equation convergence theorem.

Statement

Let $I \subseteq R$ be open, and let $u : R \times I \to R$ be six times continuously differentiable and $2 π$ -periodic in its spatial variable $x$ . Assume the nondimensional scalar wave equation $u_{t t} = u_{x x}$ on $R \times I$ . For $N \geq 3$ , $h = 2 π / N$ , $k > 0$ , $x_{j} = j h$ , and an integer $n$ with $t_{n} = n k$ and $[t_{n} - k, t_{n} + k] \subset I$ , put

U_{j}^{n} : = u (x_{j}, t_{n}), τ_{j}^{n} : = \frac{U_{j}^{n + 1} - 2 U_{j}^{n} + U_{j}^{n - 1}}{k^{2}} - \frac{U_{j + 1}^{n} - 2 U_{j}^{n} + U_{j - 1}^{n}}{h^{2}} .

Set $M_{n}$ to be the supremum

M_{n} : = \sup {∣ \partial_{x}^{4} u (x, t) ∣ : 0 \leq x \leq 2 π, t_{n} - k \leq t \leq t_{n} + k} .

Then for every grid index $j$ ,

∣ τ_{j}^{n} ∣ \leq \frac{k^{2} + h^{2}}{12} M_{n} .

Thus the normalized local truncation residual is $O (k^{2} + h^{2})$ on each compact time window where the displayed fourth-derivative bound is uniform. This consistency estimate alone is not a convergence theorem.

Statement references

A finite periodic scalar-wave benchmark

A uniform derivative bound gives a uniform Taylor remainder bound

Clairaut--Schwarz theorem for continuous second partial derivatives

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value

Absolute value in an ordered field

The triangle inequality

Facts & Assumptions

Given: A $C^{6}$ periodic solution of $u_{t t} = u_{x x}$ on the stated open time interval and the periodic grid of the scalar benchmark.

[F1]

A finite periodic scalar-wave benchmark: $h = 2 π / N$ , $x_{j} = j h$ modulo the period, and the normalized discrete second difference is the centered three-point expression in the Statement.

[F2]

A uniform derivative bound gives a uniform Taylor remainder bound: if the fourth derivative of a scalar function has absolute value at most $M$ between $x$ and $x + d$ , its Taylor remainder after degree three has absolute value at most $M ∣ d ∣^{4} / 4!$ .

[F3]

Clairaut--Schwarz theorem for continuous second partial derivatives: continuous mixed partials of the $C^{6}$ function commute, so differentiating the wave equation twice gives $u_{t t t t} = u_{x x x x}$ .

[F4]

[F5]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real-valued function on a nonempty compact metric space is bounded (and attains its extrema).

[F6]

Absolute value in an ordered field: for real $x$ , $∣ x ∣ = x$ when $x \geq 0$ and $∣ x ∣ = - x$ when $x < 0$ .

[F7]

The triangle inequality: for real $a, b$ , $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ .

Proof

technique · direct Taylor estimate

1.1givenF4F5algebra

The set $[0, 2 π] \times [t_{n} - k, t_{n} + k]$ is a nonempty closed box in $R^{2}$ : $k > 0$ and the time interval lies in $I$ . Thus [F4] makes it compact. The function $(x, t) \mapsto u_{x x x x} (x, t)$ is continuous there because $u \in C^{6}$ ; [F5] bounds it, so $M_{n} < \infty$ as claimed.

1.2F1F2F6F7givenalgebra

Apply [F2] to the spatial function $x \mapsto u (x, t_{n})$ at $x_{j}$ with increments $h$ and $- h$ . Adding the two degree-three Taylor polynomials and subtracting twice their common center value cancels their odd terms and leaves $h^{2} u_{x x} (x_{j}, t_{n})$ . By [F2], each remainder has absolute value at most $M_{n} h^{4} / 4!$ ; by [F7], the sum of the two remainders has absolute value at most $2 M_{n} h^{4} / 4!$ . After division by $h^{2}$ ,

∣ \frac{U_{j + 1}^{n} - 2 U_{j}^{n} + U_{j - 1}^{n}}{h^{2}} - u_{x x} (x_{j}, t_{n}) ∣ \leq \frac{h^{2}}{12} M_{n} .

Periodicity of $u$ and its derivatives extends the bound defining $M_{n}$ from $[0, 2 π]$ to all real spatial coordinates, including a wrapped neighbor.

1.3givenF2F3algebra

Apply the same estimate to $t \mapsto u (x_{j}, t)$ with increments $k$ and $- k$ . By [F3] and $u_{t t} = u_{x x}$ , the fourth time derivative equals $u_{t t t t} = u_{x x x x}$ , whose absolute value is bounded by $M_{n}$ on this time window and all $x$ . Thus

∣ \frac{U_{j}^{n + 1} - 2 U_{j}^{n} + U_{j}^{n - 1}}{k^{2}} - u_{t t} (x_{j}, t_{n}) ∣ \leq \frac{k^{2}}{12} M_{n} .

2.1F7step 1.2step 1.3givenalgebra∎

Subtract the two estimates of steps 1.2 and 1.3, use $u_{t t} (x_{j}, t_{n}) = u_{x x} (x_{j}, t_{n})$ , and apply the real triangle inequality [F7] to their error terms. The result is $∣ τ_{j}^{n} ∣ \leq (k^{2} + h^{2}) M_{n} / 12$ , as claimed.

Second-order local residual of the periodic wave benchmark

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources