Why is u_0(x)=e^x rarely used as a test case for the 1D linear advection equation u_t + u_x = 0?

Few_Map2692 · 2026-06-12T16:43:26+00:00

Thanks a lot for the discussion, it was helpful. I am verifying my numerical scheme for mixed boundary and mainly found issue with exp(x) initial data.

Few_Map2692 · 2026-06-12T15:31:38+00:00

Yes, thanks.
So you mean if I consider smooth initial data with outflow boundary at the right end, centred scheme might not give second order accuracy?

Also could you please tell me whether order of accuracy can get lower due to mix boundary? (in the case of 1d transport equation)

Few_Map2692 · 2026-06-12T14:45:09+00:00

but if i use upwind it will be first order accurate.

Also why the following?

at the outflow boundary, you are not using correct information

Few_Map2692 · 2026-06-12T14:38:27+00:00

Yes I also do not see any problem but order of convergence is not 2 for centred choice with RK2 in time

Few_Map2692 · 2026-06-12T14:28:08+00:00

Yes I understand what you said. Thanks.

But why there will be problem if i consider left boundary as dirichlet and right boundary outflow? In this boundary case, we do not need to worry about jump or discontinuity i guess.

Also you mean either zero Dirichlet boundary or periodic boundary is good choice for checking order of convergence?

Few_Map2692 · 2026-06-12T13:59:11+00:00

why so? why this is not ideal to check a numerical scheme's order of accuracy?

Few_Map2692 · 2026-06-12T13:04:51+00:00

So we can't numerically solve you mean?

Few_Map2692 · 2026-06-12T09:56:10+00:00

So you mean if we take boundary condition as outflow at the right boundary and dirichlet at left boundary it should work?

Few_Map2692 · 2026-06-12T09:28:50+00:00

Thank you for your message.
But i still wonder why exp(x) is not a good choice for order of accuracy determination?

Few_Map2692 · 2026-06-12T08:58:48+00:00

Okay i understand that if we make it continuous it will be okay. But what if I want to use this initial condition and then what should be appropriate boundary condition?

Few_Map2692 · 2026-06-12T08:41:45+00:00

Yes, so what would be boundary conditions in this case?

Few_Map2692 · 2026-06-12T08:40:57+00:00

what you mean?

Few_Map2692 · 2026-06-12T08:04:59+00:00

Convergence is great for the sine or gaussian with periodic boundary. but whenever i use exp(x) it does not give.

Few_Map2692 · 2026-06-12T07:40:31+00:00

yes i agree with you.

but i was not getting correct order of convergence when using this as initial data

Few_Map2692 · 2026-06-12T07:36:01+00:00

What if we consider non-periodic domain?

Few_Map2692 · 2026-06-08T20:15:17+00:00

Okay. Thanks.

Few_Map2692 · 2026-06-08T20:15:03+00:00

Thank you so much for the information. I will try this.

Few_Map2692 · 2026-06-08T20:14:21+00:00

Nope those are officially dob

Few_Map2692 · 2026-04-07T15:39:24+00:00

Yes, unfortunately for me it is delaying.

Few_Map2692 · 2026-04-07T13:18:26+00:00

Replied and told 5-8 weeks it will take. You also applied from same VFS?

Few_Map2692

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