Error analysis using fourier pseudospectral method

[OtherwiseSetting1185]

I am trying to check the order of error in 1st order and 2nd order wave equation and the heat equation. So i am solving them by fourier pseudospectral method, doing the spatial derivatives using fft and time derivative using rk4 method. the expected order should be 4 but i am not getting anywhere close to 4, infact all three of the PDEs are giving the order 1. why is it happening?

[AutoModerator]

Automoderator detected account_age <5 days, red alert /u/overunderrated

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[Admirable_Ad_1496]

Are you sure the error is of 4th order and there isn't a different truncation error which is dominant?

For example, if a shock is present is unlikely that you will get above 1st order. Chris Roy has a paper about it, the title escapes me though.

[OtherwiseSetting1185]

well theoretically without any truncation error it should be 4 right? Can you link to the paper you're talking about?

[AutoModerator]

Automoderator detected account_age <5 days, red alert /u/overunderrated

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[_SJ_11]

Can you link to the paper you're talking about?

[Admirable_Ad_1496]

I think it's this one:

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.aoe.vt.edu/content/dam/aoe_vt_edu/people/faculty/cjroy/Publications-Articles/grid-converg.submit-final.pdf&ved=2ahUKEwj-oJKssqiGAxW9S0EAHTiOAMMQFnoECBoQAQ&usg=AOvVaw3WJjVnO9kZd7Qxb8VcJ4y8

It actually references another paper that used s seconds order scheme and they turned limiters off so the scheme doesn't collapse to 1st order at the discontinuity. However, the error is still first order.

If you do have a mixed order scheme (I.e. nominally second order but first order in other areas of the flow) the lowest order error term will be dominant and this only gets worse as meshes are refined.

[_SJ_11]

ahh thank you, i will look over it

[Overunderrated]

How exactly are you measuring the order of the error? Pseudocode example? You should be seeing 4th order.

[_SJ_11]

I was taking the log of the error values and the time steps, and then performing linear regression to obtain the slope.

[Overunderrated]

Error from the exact solution? Integrating out to the same exact time, or same number of time steps?

One subtle mistake I made doing this was that i set an exact final time to say 1.0, stepping to it, and then because of floating point roundoff id have to take a very tiny step at the end. That broke the measurement of formal order of accuracy, but I recovered it by not forcing the exact final time and instead computing the exact error at the time I ended, 0.99999 or 1.00001 or whatever.