# 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

• Compressible gas – mass and momentum

• Source terms

• Diffusion

### 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)

### FVMHP03 - Linearization of Nonlinear Systems¶

Material from FVMHP Chap. 2

• General form, Jacobian matrix

• Scalar Burgers’ equation

• Compressible gas dynamics

• Linear acoustics equations

### FVMHP04 - Linear Hyperbolic Systems¶

Material from FVMHP Chap. 3

• General form, coefficient matrix, hyperbolicity

• Linear acoustics equations

• Eigen decomposition

• Characteristics and general solution

• Boundary conditions

### FVMHP05 - Linear Systems - Riemann Problems¶

Material from FVMHP Chap. 3

• Riemann problems

• Riemann problem for acoustics

• Phase plane

### 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)

### 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

• REA Algorithm

• Godunov applied to acoustics

### 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

### FVMHP09 - Dissipation, Dispersion, Modified Equations¶

Material from FVMHP Chap. 4

• Upwind, Lax-Friedrichs

• Lax-Wendroff and Beam-Warming

• Numerical dissipation and dispersion

• Modified equations

### 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

### 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

### 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

### FVMHP13 - Nonlinear scalar rarefaction waves¶

Material from FVMHP Chap. 11

• Form of centered rarefaction wave

• Non-uniqueness of weak solutions

• Entropy conditions

### 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

### FVMHP15 - Solutions and Entropy Functions¶

Material from FVMHP Chap. 12

• Weak solutions and conservation form

• Entropy functions

• Weak form of entropy condition

• Relation to vanishing viscosity solution

### 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

### 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

### 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

### 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

### 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

### 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

### FVMHP22 - Multidimensional hyperbolic problems¶

Material from FVMHP Chap. 18

• Derivation of conservation law

• Hyperbolicity

• Gas dynamics and acoustics

• Shear waves

### 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}\$

### 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

### 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