Applying Harten's TVD on Beam warming scheme

cfdnewb123 · January 26, 2025, 20:21

Consider a scheme of the form
$u^{n+1}_j=u^n_j+D_{j+1/2}\Delta u^n_{j+1/2} - C_{j-1/2} \Delta u^n_{j-1/2}$

where the scheme is TVD if
$C_{j+1/2}\geq 0, D_{j+1/2}\geq 0, C_{j+1/2}+D_{j+1/2}\leq 1$

This is straightforward for upwind or Lax Wendroff scheme. However, what about Beam-Warming scheme as it has $u^n_{j-2}$ instead of $u^n_{j+1}$ ?
Beam-Warming scheme can be expressed as follows
$u^{n+1}_j=u^n_j+(-\frac{\sigma^2}{2}+\frac{\sigma}{2})(u^n_{j-1}-u^n_{j-2}) + (0.5\sigma^2-1.5\sigma)(u^n_j-u^n_{j-1})$

FMDenaro · January 27, 2025, 14:31

Quote:

Originally Posted by cfdnewb123

Consider a scheme of the form
$u^{n+1}_j=u^n_j+D_{j+1/2}\Delta u^n_{j+1/2} - C_{j-1/2} \Delta u^n_{j-1/2}$

where the scheme is TVD if
$C_{j+1/2}\geq 0, D_{j+1/2}\geq 0, C_{j+1/2}+D_{j+1/2}\leq 1$

This is straightforward for upwind or Lax Wendroff scheme. However, what about Beam-Warming scheme as it has $u^n_{j-2}$ instead of $u^n_{j+1}$ ?
Beam-Warming scheme can be expressed as follows
$u^{n+1}_j=u^n_j+(-\frac{\sigma^2}{2}+\frac{\sigma}{2})(u^n_{j-1}-u^n_{j-2}) + (0.5\sigma^2-1.5\sigma)(u^n_j-u^n_{j-1})$

What is exactly your question, how to make BW scheme (since it is linear, the Godunov theorem says is not monotonic) non-oscillating?

cfdnewb123 · January 27, 2025, 15:50

Quote:

Originally Posted by FMDenaro

What is exactly your question, how to make BW scheme (since it is linear, the Godunov theorem says is not monotonic) non-oscillating?

I am trying to prove that BW scheme is not TVD by applying Harten's theorem. However, Harten's theorem is based off $D_{j+1/2}\Delta u^n_{j+1/2}$ and $C_{j-1/2} \Delta u^n_{j-1/2}$ , I am wondering if I can apply it to $D_{j-1/2}\Delta u^n_{j-1/2}$ and $C_{j-3/2} \Delta u^n_{j-3/2}$ instead since BW scheme has $u^n_{j-2}$ instead of $u^n_{j+1}$ ?

FMDenaro · January 27, 2025, 16:43

Quote:

Originally Posted by cfdnewb123

I am trying to prove that BW scheme is not TVD by applying Harten's theorem. However, Harten's theorem is based off $D_{j+1/2}\Delta u^n_{j+1/2}$ and $C_{j-1/2} \Delta u^n_{j-1/2}$ , I am wondering if I can apply it to $D_{j-1/2}\Delta u^n_{j-1/2}$ and $C_{j-3/2} \Delta u^n_{j-3/2}$ instead since BW scheme has $u^n_{j-2}$ instead of $u^n_{j+1}$ ?

First, I suggest to use the FV-based expression where the numerical flux function is expressed.
The Leveque textbook analyses the BW scheme and the TVD property when the limiter is introduced. You will find the details.

cfdnewb123 · January 27, 2025, 19:29

Quote:

Originally Posted by FMDenaro

First, I suggest to use the FV-based expression where the numerical flux function is expressed.
The Leveque textbook analyses the BW scheme and the TVD property when the limiter is introduced. You will find the details.

Are you referring to re-writing the above to FV-based expression such that the BW flux is
$F_{j+1/2}=au_j+\phi \frac{a}{2}(1-\sigma) (u^n_{j+1}-u^n_j)$
whereby $\phi(r)=r$ for BW flux will exceed the TVD bounds on the Sweby TVD diagram which is based on re-writing the above FDM such that $D_{j-1/2}=0$ and $0\leq C_{j-1/2}\leq 1$ .

I am wondering if I can just apply Harten's theorem without re-writing the BW scheme as its FV-based expression.

FMDenaro · January 28, 2025, 04:54

Quote:

Originally Posted by cfdnewb123

Are you referring to re-writing the above to FV-based expression such that the BW flux is
$F_{j+1/2}=au_j+\phi \frac{a}{2}(1-\sigma) (u^n_{j+1}-u^n_j)$
whereby $\phi(r)=r$ for BW flux will exceed the TVD bounds on the Sweby TVD diagram which is based on re-writing the above FDM such that $D_{j-1/2}=0$ and $0\leq C_{j-1/2}\leq 1$ .

I am wondering if I can just apply Harten's theorem without re-writing the BW scheme as its FV-based expression.

I never tried your way to demonstrate the lack of TVD property for the linear BW, furthermore I dont remember a textbook where this scheme is analysed.

If you read pag 113 of Morton and Mayers textbook, you will see the demonstration for upwind and Lax Wendroff in a FV manner.

praveen · January 31, 2025, 11:19

The incremental form is unique only for 3-point scheme involving j-1,j,j+1.

Otherwise, it is not unique, and it only gives a sufficient condition for TVD.

If you cannot verify Harten's conditions, it does not mean that the scheme is not TVD. Maybe you can satisfy them if you used a different incremental form.

So harten's criterion is useful if you want to prove a scheme is TVD.

If you want to prove it is NOT TVD, then it is not useful.

For a linear scheme of the form

$u_j^{n+1} = \sum_k c_k u_{j+k}^n$

we have

TVD <==> monotonicity preserving <==> all c_k >= 0

So to prove a linear scheme is not TVD, just show that there are some negative coefficients.

January 26, 2025, 20:21	Applying Harten's TVD on Beam warming scheme	#1
cfdnewb123 Member Join Date: Feb 2019 Posts: 72 Rep Power: 8	Consider a scheme of the form $u^{n+1}_j=u^n_j+D_{j+1/2}\Delta u^n_{j+1/2} - C_{j-1/2} \Delta u^n_{j-1/2}$ where the scheme is TVD if $C_{j+1/2}\geq 0, D_{j+1/2}\geq 0, C_{j+1/2}+D_{j+1/2}\leq 1$ This is straightforward for upwind or Lax Wendroff scheme. However, what about Beam-Warming scheme as it has $u^n_{j-2}$ instead of $u^n_{j+1}$ ? Beam-Warming scheme can be expressed as follows $u^{n+1}_j=u^n_j+(-\frac{\sigma^2}{2}+\frac{\sigma}{2})(u^n_{j-1}-u^n_{j-2}) + (0.5\sigma^2-1.5\sigma)(u^n_j-u^n_{j-1})$

January 31, 2025, 11:19		#7
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 345 Blog Entries: 6 Rep Power: 19	The incremental form is unique only for 3-point scheme involving j-1,j,j+1. Otherwise, it is not unique, and it only gives a sufficient condition for TVD. If you cannot verify Harten's conditions, it does not mean that the scheme is not TVD. Maybe you can satisfy them if you used a different incremental form. So harten's criterion is useful if you want to prove a scheme is TVD. If you want to prove it is NOT TVD, then it is not useful. For a linear scheme of the form $u_j^{n+1} = \sum_k c_k u_{j+k}^n$ we have TVD <==> monotonicity preserving <==> all c_k >= 0 So to prove a linear scheme is not TVD, just show that there are some negative coefficients. __________________ http://cpraveen.github.io http://twitter.com/cfdlab http://github.com/cpraveen

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