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Phi functions at zero require no matrix division

Statement

For $A = 0$ , $g (t, y) = t$ , $y (0) = 0$ , and a step $h > 0$ , compute the exponential-Euler and ETD2 steps directly from the series values $φ_{1} (0) = 1$ and $φ_{2} (0) = 1 / 2$ . The ETD2 step is exact, while exponential Euler returns zero.

Facts & Assumptions

Given: The scalar IVP and exact method maps of Exponential Euler and the two-stage ETD2 step maps.

[F1]

The phi-function family by everywhere-convergent series and The scalar and matrix phi series converge absolutely: at zero, $φ_{1} (0) = 1$ , $φ_{2} (0) = 1 / 2$ , and $e^{0} = 1$ , directly from the convergent series.

[F2]

Exponential Euler and the two-stage ETD2 step maps: the ETD2 predictor is exponential Euler and its correction uses $φ_{1} - φ_{2}$ and $φ_{2}$ .

[F3]

First-order systems, initial value problems, and solutions on intervals and If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz: a solution of $y^{'} = t$ is differentiable and its endpoint change is the integral of its derivative.

Verification

1.1givenF3algebra

By [F3], the exact solution is $y (t) = \int_{0}^{t} s d s = t^{2} / 2$ , so $y (h) = h^{2} / 2$ .

2.1F1F2step 1.1algebra

Since $A = 0$ , $e^{h A} = 1$ and $g (0, y (0)) = 0$ . The exponential-Euler step is therefore $0 + h φ_{1} (0) 0 = 0$ , and its predictor for ETD2 is also $a = 0$ .

3.1F1F2step 1.1step 2.1algebra∎

At the ETD2 endpoint, $g (h, a) = h$ . Its update is $0 + h (φ_{1} (0) - φ_{2} (0)) 0 + h φ_{2} (0) h = h^{2} / 2$ , which equals the exact value from step 1.1. This calculation evaluates $φ_{k} (0)$ from the series and never divides by $A$ or its eigenvalues.

Depends on

Used by

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Sources

Jitse Niesen and Will M. Wright, A Krylov Subspace Algorithm for Evaluating the Phi-Functions Appearing in Exponential Integrators, arXiv:0907.4631v2 (standard reference, not scraped)
Marlis Hochbruck and Alexander Ostermann, Exponential Integrators, Acta Numerica 19 (2010) (standard reference, not scraped)