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[diff] L2 method for Caputo derivatives is first order #2

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alexfikl opened this issue May 7, 2023 · 1 comment
Open

[diff] L2 method for Caputo derivatives is first order #2

alexfikl opened this issue May 7, 2023 · 1 comment
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@alexfikl
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alexfikl commented May 7, 2023

The L2C method seems to be working correctly. Likely just an index issue or something? Needs a bit of debugging.

@alexfikl alexfikl changed the title Diff: L2 method for Caputo derivatives is first order [diff] L2 method for Caputo derivatives is first order May 7, 2023
@alexfikl alexfikl added the bug Something isn't working label May 7, 2023
@alexfikl
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alexfikl commented Sep 18, 2023

The L2 method applied to the Caputo derivative with $\alpha = 1.5$ looks like

h Error EOC
3.333e-02 6.659753e-02 ---
1.613e-02 3.341454e-02 0.950
7.937e-03 1.690615e-02 0.961
3.937e-03 8.552690e-03 0.972
1.961e-03 4.318041e-03 0.980
9.785e-04 2.175312e-03 0.986
6.519e-04 1.455290e-03 0.990
4.888e-04 1.093799e-03 0.991
Overall 0.975

The L2 method applied to the Riemann-Liouville derivative with $\alpha = 1.5$ (through D_{RL} = D_C - F(a)) gives

h Error EOC
3.333e-02 3.686930e-02 ---
1.613e-02 9.678625e-03 1.842
7.937e-03 2.593723e-03 1.857
3.937e-03 6.847805e-04 1.900
1.961e-03 1.774681e-04 1.937
9.785e-04 4.536133e-05 1.962
6.519e-04 2.033538e-05 1.976
4.888e-04 1.149314e-05 1.981
Overall 1.916

For the Caputo derivative, it's first-order regardless of $\alpha$, but for the RL derivative it seems to behave roughly like $\alpha + 0.5$ instead of the expected $3 - \alpha$. Note quite sure what to make of that.

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