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A periodic manufactured solution for the diagonal-diffusion HJB stencil

Statement

For $d = 1$ , one control, $α = 1$ , and $b = 0$ on $T^{1}$ , set $u (x, t) = e^{- t} \cos (2 π x), f (x, t) = (- 1 + 4 π^{2}) e^{- t} \cos (2 π x) .$ Then $u_{t} - (u_{x x} + f) = 0$ exactly. On a periodic grid $h = 1 / N$ , the centered diffusion scheme with $0 < τ \leq h^{2} / 2$ satisfies a normalized one-step residual of order $O (τ + h^{2})$ on the sampled exact solution. This is a local residual calculation, not a global solution-error estimate.

Facts & Assumptions

Given: The smooth periodic function and forcing in the statement and the one-control explicit scheme of Explicit monotone diagonal-diffusion Bellman grid update.

[F1]

The continuous operator in one dimension is $u_{t} - (α u_{x x} + b u_{x} + f) = 0$ (Periodic diagonal controlled-diffusion HJB Cauchy problem).

[F2]

Under $b = 0$ and $α = 1$ , the normalized scheme residual equals the time difference quotient minus $D_{x x}^{h} u + f$ ; its second-difference error on smooth tests is controlled by Smooth consistency of the diagonal-diffusion Bellman stencil.

Proof

1.1F1givenalgebra

Direct differentiation gives $u_{t} = - u$ and $u_{x x} = - 4 π^{2} u$ . Therefore $u_{t} - u_{x x} = (- 1 + 4 π^{2}) u = f$ , proving the exact PDE identity in [F1]. The diffusion-dominance condition is automatic since $b = 0$ ; the CFL condition is $2 τ / h^{2} \leq 1$ , which is exactly the stated time-step bound.

2.1F2step 1.1algebra∎

The discrete second difference is $D_{x x}^{h} u (x_{i}, t_{n}) = - \frac{4 \sin^{2} (π h)}{h^{2}} u (x_{i}, t_{n}) .$ Taylor expansion in time gives $(u (x_{i}, t_{n} + τ) - u (x_{i}, t_{n})) / τ = u_{t} (x_{i}, t_{n}) + O (τ)$ , while $u_{x x} - D_{x x}^{h} u = (\frac{4 \sin^{2} (π h)}{h^{2}} - 4 π^{2}) u = O (h^{2}),$ because $\sin z = z + O (z^{3})$ at $z = 0$ . Thus the exact normalized residual is $O (τ + h^{2})$ , agreeing with [F2]. No accumulated error is claimed.

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Sources

Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations (standard reference, not scraped)
Guy Barles and Panagiotis E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations (standard reference, not scraped)