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On the Burgers equation with moving boundary

Asymptotic solutions for rapidly oscillating boundary.

Analytical Solutions for the Unsteady Compressible Flow Equations Serving as Test Cases for the Verification of Numerical Schemes.pdf

From Equation (68) downwards we find a transformation for the burgers equation into the heat equation.

This would allow us to verify our solutions.

What is it?
How is it carried out into the equations?

References:

• http://oomph-lib.maths.man.ac.uk/doc/unsteady_heat/two_d_unsteady_heat_ALE/html/index.html

• PB: Product Backlog
• FOM: Full Order Model
• ROM: Reduced Order Model
• DEIM: Discrete Empirical Interpolation Method
• MDEIM: Matrix Discrete Empirical Interpolation Method
• TH: Theoretical Analysis
• LT: Literature Review

References:

• https://www.amherst.edu/system/files/latextips.pdf
• https://www.math.tamu.edu/~harold.boas/courses/math696/math-macros.html

See content of 99aa815.

Right now finding a relevant paper and including it in my references database is a slow process.

Making this approach faster could help me digitalize my thoughts on each reading.

Related #11.

• Generic FOM definition.
• Method of Manufactured Solutions.
• Generic ROM definition: Reduced Basis Method, from local to global support.
• Basis construction (POD).
• Lifting function (#3, #7)
• ROM integration.
• FOM reconstruction.

• (20%) Literature Review: Research questions and related works #5 #10 #11.
• (50%) FOM, ROM and HROM definition and presentation #22 #20.
• Problem presentation:
  • One-Dimensional Burger's Equation
  • Oscillating cylinder - 2D Navier-Stokes equations.
  • Beating Heart - 2D Navier-Stokes equations.
• Appendices:
  • Manufactured Problems: Steady-State and Transient.
  • Geometrical Conservation Laws: Constant solution #24 #23.

Use this paper to see how they check convergence rates for BDF-2.

https://arxiv.org/pdf/1907.00406.pdf

• Higher-order time marching scheme, BD2.
• Comparison with generalized transformation from a moving to a fixed domain. #22
• How to compare different problems.
• Relative efficiency of the reduction technique.

Father: #8

An Introduction to Fluid-Structure Interaction: Application to the Piston Problem

keywords: moving piston, FSI, ALE, fluid-structure interaction, mass-spring dynamics, one-dimensional fluid flow, finite ele- ments, energy conservations, mesh deformation

• Very nice and detailed examination of the 1D piston problem for the compressible NS equations.
• Includes demonstration of the unphysical effects present when the ALE term is not included (pure Eulerian approach).

borggaard2011avp.pdf

Abstract

We introduce improved reduced-order models for turbulent flows. These models are inspired from successful methodologies used in large eddy simulation, such as artificial viscosity, applied to standard models created by proper orthogonal decomposition of flows coupled with Galerkin projection. As a first step in the analysis and testing of our new methodology, we use the Burgers equation with a small diffusion parameter. We present a thorough numerical analysis for the time discretization of the new models. We then test these models in two problems displaying shock-like phenomena. Of course, since the Burgers equation does not model turbulence, we next need to test our new models in realistic turbulent flow settings. This is the subject of a forthcoming report.

What is this?
Why is it relevant?

It seems to relate the ability to reproduce unsteady constant solutions in moving domains.
In other words: how we discretize the deformation of the domain, is it exact, or does it act like a source or sink?

The following transient problem:

u_t - 0.01 \Delta u = 0
u_0 = 1
u_D = 1

should remain u(x,t) = 1 if the discretisation does not introduce any time-marching errors.

• Fixed domain formulation.
• Moving domain formulation (ALE).

https://arxiv.org/abs/1607.05622

P L Sachdev - Nonlinear diffusive waves-Cambridge University Press (1987).pdf

• Contains derivation for the isentropic Euler equations which end up with a one-dimensional nonlinear Burger's-like equation which presumably retains.

Define implementation details for lifting function term.

KikeM/msc-thesis#12

References:

• https://tug.org/tug2019/slides/slides-ziegenhagen-python.pdf

Extensive document to hold the implementation details.

A subset of sections and paragraphs from such document should actually be the Thesis manuscript.

Motivation
From working at ETS, I have realized that it is healthy to have a place where the equations in their long form, or important details, are centralized.
As projects grow, it is easy to lose sight or memory of important stuff.
Then, the final thesis document should be a cut-out version of this document.

Using the technology introduced by subfiles, I want to set up a master document where I can keep track of all the details and then simply compile certain sections together to get a brief thesis manuscript.

The idea is that I have master document where I dump everything, so that I have a clean a detailed document of my definitions and concerns, but still be able to produce a clean, neat and short thesis document.

• https://www.overleaf.com/learn/latex/Management_in_a_large_project
• https://www.overleaf.com/learn/latex/Multi-file_LaTeX_projects
• https://tex.stackexchange.com/questions/7052/how-do-i-choose-which-sections-to-compile-from-a-latex-document

quasi-linear-parabolic-aerodynamics.pdf

url: https://www.jstor.org/stable/43633894

keywords:

@article{10.2307/43633894,
 ISSN = {0033569X, 15524485},
 URL = {http://www.jstor.org/stable/43633894},
 author = {JULIAN D. COLE},
 journal = {Quarterly of Applied Mathematics},
 number = {3},
 pages = {225--236},
 publisher = {Brown University},
 title = {ON A QUASI-LINEAR PARABOLIC EQUATION OCCURRING IN AERODYNAMICS},
 volume = {9},
 year = {1951}
}

Very nice explanation of the meaning of the Burgers equation.

OnSomeApproximateExactSolutionsBoundaryValueProblemsBurgersEquation.pdf

Keywords: moving piston, burgers equation

Contains analytical works and approximation of solutions.

The moving piston they consider is placed at x=0, and the domain is infinite, so the waves propagate on both directions and never interact.

This could be a good departing point, my domain is finite but I would be setting an outflow condition.

A TABLE OF SOLUTIONS OF THE ONE-DIMENSIONAL BURGERS EQUATION

Literature survey

Piston references: 2, 34, 52, 65

Through the jacobian transformation, how can we solve the heat equation in a moving domain using a reference fixed domain.

Reference : https://www.oden.utexas.edu/media/reports/2003/0325.pdf

Lattice Boltzmann method for fluid flow around bodies using volume penalization

Simplified FSI problem: couple burgers' equation with spring-like ODE for boundary motion.

Problem to address: uncentralized knowledge.

Facts

• I have named the papers with the following structure: YYYY_camelCaseTitle_aAuthor.
• I have classified them in a couple of folders, namely POD, RB, FEM, EIM.
• I read a paper:
  • Learn something.
  • Come up with questions.
  • Reach conclusions.
  • Find new references to read.
• I take notes on the printed paper, because I feel this helps me internalize better my thoughts on it.

Problem

• Gained knowledged now remains stuck in the physical format of the paper.
  This ain't bad, cause my memory works better that way.
• I find it difficult to put together related topics/methodologies/conclusions.

Solution (?)

• Read a paper in the traditional way, then digitalize conclusions, insights.
• For each paper, if relevant, transcribe their formulation to ours (see #9).
  This will ease comparison across references.
• Set up an easy to use folder structure to include this digitalization into the latex documents (see #11).

References:

• https://libguides.umn.edu/systematicreviews
• https://libguides.csu.edu.au/systematicreviews/question

Related to #20

Solution of the transport equations using a moving coordinate system.pdf

reference: #33

Explains how solving a PDE in a moving coordinate domain is equivalent to having the boundary conditions change location!

Explain how the lifting is implemented in a 3D setting.

Facts:

• Inhomogeneous boundary conditions spoil the construction of the basis.
• The deforming boundary has homogeneous boundary conditions.
• The lifting only needs to satisfy the boundary conditions and to be sufficiently smooth, it doesn't need to be perfect.

Reference: #10

Three points to cover:

1. Needs.
2. What others have done.
3. What we intend to do:
   • SMART research questions.