Lecture videos and slides

25 video lectures, most of which are 30-50 minutes long, were recorded as supplemental materials for this class. They are available on the Clawpack YouTube Channel in this playlist. Each video is also linked individually below.

Slides for each lecture (pdf files) are linked below and can also be found in the slides directory of the Class GitHub Repository, or in the Github repository clawpack/fvmhp_materials, which also includes the latex files that created the slides.

Lecture contents

“FVMHP” refers to the Textbook.

FVMHP01 - Derivation of Conservation Laws

Material from FVMHP Chap. 2

  • Integral form in one space dimension

  • Advection

  • Compressible gas – mass and momentum

  • Source terms

  • Diffusion

SlidesYouTube video

FVMHP02 - Variable Coefficient Advection

Material from FVMHP Sec. 9.1

  • Quasi-1D pipe

  • Units in one space dimension

  • Conservative form: $q_t + (u(x)q)_x = 0$

  • Advective form: $q_t + u(x)q_x = 0$ (color equation)

SlidesYouTube video

FVMHP03 - Linearization of Nonlinear Systems

Material from FVMHP Chap. 2

  • General form, Jacobian matrix

  • Scalar Burgers’ equation

  • Compressible gas dynamics

  • Linear acoustics equations

SlidesYouTube video

FVMHP04 - Linear Hyperbolic Systems

Material from FVMHP Chap. 3

  • General form, coefficient matrix, hyperbolicity

  • Scalar advection equation

  • Linear acoustics equations

  • Eigen decomposition

  • Characteristics and general solution

  • Boundary conditions

SlidesYouTube video

FVMHP05 - Linear Systems - Riemann Problems

Material from FVMHP Chap. 3

  • Riemann problems

  • Riemann problem for advection

  • Riemann problem for acoustics

  • Phase plane

SlidesYouTube video

FVMHP06 - Linear Systems - Nonhyperbolic Cases

Material from FVMHP Chap. 3, 16

  • Acoustics equations if $K_0 < 0$ (eigenvalues complex)

  • Acoustics equations if $K_0 = 0$ (not diagonalizable)

  • Coupled advection equations

SlidesYouTube video

FVMHP07 - Introduction to Finite Volume Methods

Material from FVMHP Chap. 4

  • Comparsion to finite differences

  • Conservation form, importance for shocks

  • Godunov’s method, wave propagation view

  • Upwind for advection

  • REA Algorithm

  • Godunov applied to acoustics

SlidesYouTube video

FVMHP08 - Accuracy, Consistency, Stability, CFL Condition

Material from FVMHP Chap. 4, 8

  • Order of accuracy, local and global error

  • Consistent numerical flux functions

  • Stability

  • CFL Condition

SlidesYouTube video

FVMHP09 - Dissipation, Dispersion, Modified Equations

Material from FVMHP Chap. 4

  • Upwind, Lax-Friedrichs

  • Lax-Wendroff and Beam-Warming

  • Numerical dissipation and dispersion

  • Modified equations

SlidesYouTube video

FVMHP10 - High-resolution TVD methods

Material from FVMHP Chap. 6

  • Godunov: wave-propagation and REA algorithms

  • Extension of REA to piecewise linear

  • Relation to Lax-Wendroff, Beam-Warming

  • Limiters and minmod

  • Monotonicity and Total Variation

Slides`YouTube video <>`__

FVMHP11 - TVD Methods and Limiters

Material from FVMHP Sec. 6.11, 6.12

  • Slope limiters vs.flux limiters

  • Total variation for scalar problems

  • Proving TVD in flux-limiter form

  • Design of TVD limiters

  • Sweby Region

SlidesYouTube video

FVMHP12 - Nonlinear Scalar PDEs, Traffic flow

Material from FVMHP Chap. 11

  • Traffic flow — car following models

  • Traffic flow — conservation law

  • Shock formation

  • Rankine-Hugoniot jump conditions

  • Riemann problems

SlidesYouTube video

FVMHP13 - Nonlinear scalar rarefaction waves

Material from FVMHP Chap. 11

  • Form of centered rarefaction wave

  • Non-uniqueness of weak solutions

  • Entropy conditions

SlidesYouTube video

FVMHP14 - Finite Volume Methods for Scalar Conservation Laws

Material from FVMHP Chap. 12

  • Godunov’s method

  • Fluxes, cell averages, and wave propagation form

  • Transonic rarefactions waves

  • Approximate Riemann solver with entropy fix

SlidesYouTube video

FVMHP15 - Solutions and Entropy Functions

Material from FVMHP Chap. 12

  • Weak solutions and conservation form

  • Admissibility / entropy conditions

  • Entropy functions

  • Weak form of entropy condition

  • Relation to vanishing viscosity solution

SlidesYouTube video

FVMHP16 - Convergence to Weak Solutions and Nonlinear Stability

Material from FVMHP Chap. 12

  • Lax-Wendroff Theorem

  • Entropy consistent finite volume methods

  • Nonlinear stability

  • Total Variation stability

SlidesYouTube video

FVMHP17 - Nonlinear systems, shock waves

Material from FVMHP Chap. 13

  • Shallow water equations

  • Rankine-Hugoniot condition

  • Hugoniot locus in phase space

  • All-shock Riemann solutions

SlidesYouTube video

FVMHP18 - Rarefaction waves and integral curves

Material from FVMHP Chap. 13

  • Integral curves

  • Genuine nonlinearity and rarefaction waves

  • General Riemann solution for shallow water

  • Riemann invariants

  • Linear degeneracy and contact discontinuities

SlidesYouTube video

FVMHP19 - Gas dynamics and Euler equations

Material from FVMHP Chap. 14

  • The Euler equations

  • Conservative vs.primitive variables

  • Contact discontinuities

  • Projecting phase space to $p$–$u$ plane

  • Hugoniot loci and integral curves

  • Solving the Riemann problem

SlidesYouTube video

FVMHP20 - Finite volume methods for nonlinear systems

Material from FVMHP Chap. 15

  • Wave propagation method for systems

  • High-resolution methods using wave limiters

  • Example for shallow water equations

SlidesYouTube video

FVMHP21 - Approximate Riemann solvers

Material from FVMHP Chap. 15

  • HLL method

  • Linearized Jacobian approach

  • Roe solvers

  • Shallow water example

  • HLLE method and positivity

  • Harten-Hyman entropy fix

SlidesYouTube video

FVMHP22 - Multidimensional hyperbolic problems

Material from FVMHP Chap. 18

  • Derivation of conservation law

  • Hyperbolicity

  • Advection

  • Gas dynamics and acoustics

  • Shear waves

SlidesYouTube video

FVMHP23 - Fractional step methods

Material from FVMHP Chap. 19

  • Dimensional splitting (Chapter 19)

  • Fractional steps for source terms (Chapter 17)

  • Godunov and Strang splitting

  • Cross-derivatives in 2D hyperbolic problems

  • Upwind splitting of $ABq_{yx}$ and $BAq_{xy}$

SlidesYouTube video

FVMHP24 - Multidimensional finite volume methods

Material from FVMHP Chap. 19–21

  • Integral form on a rectangular grid cell

  • Flux differencing form

  • Scalar advection: donor cell upwind

  • Corner transport upwind and transverse waves

  • Wave propagation algorithms for systems

  • Transverse Riemann solver

SlidesYouTube video

FVMHP25 - Acoustics in Heterogeneous Media

Material from FVMHP Chap. 9, 21

  • One space dimension

  • Reflection and transmission at interfaces

  • Non-conservative form, Riemann problems

  • Two space dimensions

  • Transverse Riemann solver

  • Some examples

SlidesYouTube video