Some questions about the theory paper of CESE method (doi:10.1006/jcph.1995.1137)

[yuhow]

Dear all,

The following message is some of my questions about the alpha-mu scheme in the theory paper of CESE method (doi:10.1006/jcph.1995.1137)

Basically, all the following questions are based on the Section 2 (alpha-mu scheme) of the paper.

1. Why does Eq. (2.4) imply Eq. (2.5).?
2. Why can Eq. (2.9) be proved using the fact that the total flux of h* leaving the boundary of any space-time region that is the union of any combination of CEs vanishes? Can't figure it out.
3. (In page 301) What's the finite-difference approximation?
4. (In page 301) What's the meaning of "the alpha-mu scheme uses a mesh that is staggered in time"?
5. (In page 301) What's the Lax scheme?
6. (In page 301) What's the amplification factors? Also what's their meaning/usage in the Leapfrog/DuFort-Frankel scheme?
7. (In page 301) What's the meaning of "two-level" and "three-level" scheme?
8. Why does not solutions of Eq. (2.22) dissipate with time? Or why is "no dissipation" equivalent to "neutrally stable"?
9. Why is the total flux leaving any conservation element zero? This question is relevant to the definition of Eq. (2.28).
10. Why does the term of second order partial derivative w.r.t. x vanish in Eq. (2.29)?
11. Eq. (2.33) maybe is wrong. The partial derivative in the left-hand side of Eq. (2.33) should be with respect to x instead of t.
12. (In page 304) Why is the local convective motion of physical variables relative to the moving mesh kept to a minimum if the space-time mesh is allowed to evolve with the physical variables?
13. (In page 304) What's the meaning of "principal" and "spurious" amplification factors?

Please help me. So many thanks in advance.

[yungyuc]

When referring to a term in a page, please also indicate which line and paragraph it is. Or, quote the context in the question. It will save reviewers' time.

1. Why does Eq. (2.4) imply Eq. (2.5).?

   Substitute Eq. (2.3) into Eq. (2.1), and you will see Eq. (2.5).

2. Why can Eq. (2.9) be proved using the fact that the total flux of h* leaving the boundary of any space-time region that is the union of any combination of CEs vanishes? Can't figure it out.

   Equation (2.9) is simply definition. It cannot be proved. It turns the surface integral into a line integral as shown in Fig. 1.

3. (In page 301) What's the finite-difference approximation?

   The definition of finite difference is the approximation method of derivatives by using Taylor's expansion. The wikipedia page is informative: https://en.wikipedia.org/wiki/Finite_difference

4. (In page 301) What's the meaning of "the alpha-mu scheme uses a mesh that is staggered in time"?

   The spatial grid at $t=t^{n+\frac{1}{2})$ is staggered to that at $t=t^n$.

5. (In page 301) What's the Lax scheme?

   Lax-Wendroff scheme. See P. Lax and B. Wendroff, “Systems of conservation laws,” Comm. Pure Appl. Math., vol. 13, no. 2, pp. 217–237, May 1960.

6. (In page 301) What's the amplification factors? Also what's their meaning/usage in the Leapfrog/DuFort-Frankel scheme?

   Amplification factor appears in stability analysis. I think you are referring to Eq. (2.21). It "looks like" a result of Von Neumann stability analysis.

7. (In page 301) What's the meaning of "two-level" and "three-level" scheme?

   At the last fourth line in the second paragraph of the right column in page 301, the "two-level explicit scheme" means the explicit scheme uses two time steps. I don't know why at the last second line of the first paragraph said the DuFort-Frankel scheme is three-level. But it's not important to developing CESE code.

8. Why does not solutions of Eq. (2.22) dissipate with time? Or why is "no dissipation" equivalent to "neutrally stable"?

   Eqation (2.22) doesn't have a dissipation (second-order) term. The paragraph of the equation is talking about an analogy rather than a theory. It takes more time to discuss this.

9. Why is the total flux leaving any conservation element zero? This question is relevant to the definition of Eq. (2.28).

   The flux conservation is our governing equations, Eqs. (2.1) and (2.2). Conservation elements and Eqs. (2.27) and (2.28) are approximation of the governing equations.

10. Why does the term of second order partial derivative w.r.t. x vanish in Eq. (2.29)?

    The approximation of $\mathbf{h}^*$ forces that. Note Eq. (2.29) is a definition.

11. Eq. (2.33) maybe is wrong. The partial derivative in the left-hand side of Eq. (2.33) should be with respect to x instead of t.

    Yes. As we discussed it seemed to be a typo. Looking at Eq. (2.34) the derivative in Eq. (2.33) should be with respect to x.

12. (In page 304) Why is the local convective motion of physical variables relative to the moving mesh kept to a minimum if the space-time mesh is allowed to evolve with the physical variables?

    Could you point out which paragraph you are referring to?

13. (In page 304) What's the meaning of "principal" and "spurious" amplification factors?

    I believe it was defined in TM 104495, as cited in bullet (b) at the last paragraph of the left column in page 304. This detail is important for numerical method design, but code development won't use it.

[yuhow]

Thanks for your reply!
Some of them are solved.
But I find that there is a type in my original Q2:

Why can "Eq. (2.19)" be proved using the fact that the total flux of h* leaving the boundary of any space-time region that is the union of any combination of CEs vanishes? Can't figure it out.

It should be Eq. (2.19). Sorry for misleading you.

And sorry for the lack of information in Q12:

(In page 304) Why is the local convective motion of physical variables relative to the moving mesh kept to a minimum if the space-time mesh is allowed to evolve with the physical variables? The relevant description is in the end of bullet (a) at the second-last paragraph of the left column in page 304.

[yungyuc]

• Q2: Is there a specific point you don't understand in the paragraph (last third in the right column in page 300). The whole paragraph is explaining that.
• Q12: I didn't do the analysis myself, but I think the statement refers to comparison to other classical methods, e.g., leapfrog. To understand this I think one should carry out the error analysis for all the methods mentioned.

[yuhow]

Few more questions from Section 5 (The Euler Solver):

• Q1: I couldn't figure out why the last paragraph of the left column in the page 310 said that one would expect that $| (u_{mx+})^n_j | >> | (u_{mx-})^n_j |$. Basically, it was not that easy for me to catch the key point of this paragraph.
• Q2: The paragraph, which was just right after Eq. (4.39) in page 310, mentioned that:
  For $\alpha > 0$, this average is biased toward the one among $x+$ and $x_-$ with the smaller magnitude. For the same value of $|x_+|$ and $|x_-|$, the bias increases as $\alpha$ increases. Thus, we should always choose $\alpha \geq 0$._
  After few hours work, I still could't understand the meaning of this description. Please help me.
• Q3: A description in the second-last paragraph of the right column in page 310 mentioned that:
  $u{mx\pm}^n_j$ are constructed using only the data associated with the mesh points (j - 1/2, n - 1/2) and (j + 1/2, n - 1/2), the effect of this modification is highly local; i.e., it generally will not cause the smearing of shock discontinuities._
  I didn't quite understand this modification is highly 'local' and will not cause the smearing of shock discontinuities.

[yungyuc]

The parts you read is about designing weighting functions for treating discontinuities. It's complex. Many of the techniques are developed with experiences and can be ad hoc. They are necessary for treating strong non-linear equations like the Euler equations, in which shocks exist.

• Q1 response: "In case a discontinuity occurs between $(j,n)$ and $(j+\frac{1}{2},n)$" means $|(u_{mx+}) _ j^n|$ (the difference in the region) is very large. Note because of the discontinuity, at a certain location in the region, the derivative is undefined, or you can expect the gradient of $u$ is infinite. On the other hand, if the region between $(j-\frac{1}{2},n)$ and $(j,n)$ is smooth (without discontinuity), the derivative is finite (and relatively small). It's reasonable to expect a finite gradient (left difference $|(u_{mx-}) _ j^n|$) much less than a value used to model an infinite gradient (right difference $|(u_{mx+}) _ j^n|$).
• Q2 response: The key of this treatment is to provide a weighted average for the value of $(u_{mx}) _ j^n$ based on $(u_{mx\pm}) _ j^n$. The central differencing formula $(u_{mx}) _ j^n = \frac{1}{2}\left((u_{mx-}) _ j^n + (u_{mx-}) _ j^n\right)$ averages (smears) too much and do not correctly capture the discontinuous solution. Equation (4.39) is one way to fix the central differencing. The same idea exists in upwind differencing.
• Q3 response: The larger the stencil of a scheme is, the more likely it suffers from smearing or overshoot. Averaging with more points smooths the solution, but bringing more high frequency components into the solution gets you more oscillation. Therefore the more local a scheme is, generally, the more resistive it is to smearing.

To more understand the discontinuity treatments, you need to experiment with various settings in a code. It becomes simple how $\alpha$ works when you play it numerically. Treating discontinuity is of utmost importance for solving non-linear PDEs. Essentially, PDEs aren't solvable at discontinuity (derivatives aren't defined). That's why semi-analytical methods like the spectral element method doesn't work for me. But for many problems, Mother Nature tells us there indeed is one, and only one solution she prefers. To this point, it should be clear to you that what the CESE method really solves isn't the differential equation $\frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} = 0$, but the integral equation $\oint \mathbf{h}\cdot\dif\mathbf{s} = 0$. By adding constraints, the CESE method approximates the DE solution (outside the discontinuity) with the IE solution that Mother Nature likely chooses.

You can read the whole theory in Courant76 and Lax73. But I suggest you to finish all the coding before reading it. The theory of conservation laws takes more time to digest than the CESE method itself.