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Global weighted L2 error from the complete parabolic residual

Statement

Under the smoothness, exact initial data, and exact sampled Dirichlet rows of Local truncation residual for the θ heat scheme, let $θ \in [1 / 2, 1]$ and let $U^{n}$ be the fully discrete solution. At every time level $m \leq M$ ,

∥ r^{m} - U^{m} ∥_{h} \leq 2 T \sqrt{L_{x} L_{y}} R_{θ},

where

R_{θ} = {\begin{cases} Δ t M_{t t} + (κ_{x} h_{x}^{2} M_{x 4} + κ_{y} h_{y}^{2} M_{y 4}) / 12, & θ \neq 1 / 2, \\ (Δ t)^{2} M_{t t t} / 12 + (κ_{x} h_{x}^{2} M_{x 4} + κ_{y} h_{y}^{2} M_{y 4}) / 12, & θ = 1 / 2. \end{cases}

Here $r^{m}$ is the exact solution sampled on the grid and $∥ \cdot ∥_{h}$ is the weighted interior L2 norm. Thus the global rate is first order in time when $θ \neq 1 / 2$ , second order in time when $θ = 1 / 2$ , and second order in each spatial spacing under the stated derivative and boundary hypotheses.

Facts & Assumptions

Given: The finite rectangular grid and time levels of Fully discrete θ heat finite-difference scheme, an exact smooth solution and exact sampled initial and Dirichlet rows as in Local truncation residual for the θ heat scheme, and $θ \in [1 / 2, 1]$ .

[F1]

Fully discrete θ heat finite-difference scheme uses the same linear θ update and the same exact boundary row at each time level for both the numerical and sampled exact grids.

[F2]

Unconditional weighted L2 stability for θ at least one half: for a zero-boundary error step with source $q^{n}$ , $∥ e^{n + 1} ∥_{h} \leq ∥ e^{n} ∥_{h} + 2 Δ t ∥ q^{n} ∥_{h}$ .

[F3]

Local truncation residual for the θ heat scheme: the exact sampled grid has complete interior residual $τ^{n}$ with $∥ τ^{n} ∥_{\infty} \leq R_{θ}$ .

Proof

1.1F1F3givenalgebra

Define the interior error $e^{n} = r^{n} - U^{n}$ . Exact initial values give $e^{0} = 0$ , and exact boundary rows give $e^{n} = 0$ on every boundary node. Subtracting the numerical equation from the exact sampled equation whose residual is [F3] shows that $e^{n}$ satisfies the same zero-Dirichlet θ update with source $q^{n} = τ^{n}$ .

2.1F2step 1.1algebra

Applying [F2] at each time step gives $∥ e^{n + 1} ∥_{h} \leq ∥ e^{n} ∥_{h} + 2 Δ t ∥ τ^{n} ∥_{h}$ . Summing these inequalities from $n = 0$ to $m - 1$ and using $e^{0} = 0$ yields $∥ e^{m} ∥_{h} \leq 2 Δ t \sum_{n = 0}^{m - 1} ∥ τ^{n} ∥_{h}$ .

3.1F3step 2.1givenalgebra∎

The weighted grid norm has total interior weight $h_{x} h_{y} (N_{x} - 1) (N_{y} - 1) < L_{x} L_{y}$ , so [F3] gives $∥ τ^{n} ∥_{h} \leq \sqrt{L_{x} L_{y}} R_{θ}$ . Since $m Δ t \leq T$ , step 2.1 implies $∥ e^{m} ∥_{h} \leq 2 T \sqrt{L_{x} L_{y}} R_{θ}$ . Substituting the two explicit values of $R_{θ}$ from [F3] proves the stated rates.

Remarks

Exact boundary samples make the error zero on the boundary; the local residual includes every interior stencil row adjacent to it. The conclusion is a smooth-solution grid L2 error estimate; it gives no positivity or nonsmooth-solution rate. No choice axiom is used.

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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)