Alphabeta MathCompressible numerical conservation and admissibility
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Conservative finite-volume weak limits
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Density Separability and Convolution in Lᵖ
- Distributions Test Functions and Differentiation
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Positivity, invariant domains and entropy-stable systems fluxes
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Regular Surfaces and Surface Integrals
- Relations, Functions, and Quotients
- Riemann solvers for hyperbolic systems
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Scalar Conservation Laws and Entropy Solutions
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Subspaces, Products, and Quotients
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Divergence Theorem and Classical Stokes
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Euler state, flux, positive set and numerical roles
Proposition
Ideal-gas LLF admissibility specialization
Proposition
Shared-face flux telescoping conservation
Remark
Entropy, invariant-domain and weak-limit hypotheses
Sources
- H. De Sterck and P. Ullrich, Introduction to Computational PDEs, Chapter 3
- E. Tadmor, Entropy Stable Schemes for Hyperbolic Conservation Laws, Handbook of Numerical Methods for Hyperbolic Problems, Vol. 17
- D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo, Riemann Problems and Jupyter Solutions
- Xiangxiong Zhang and Chi-Wang Shu, On Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Euler Equations on Rectangular Meshes
- Benjamin Seibold, MIT OpenCourseWare 18.336 Lecture 19, Finite Volume Methods