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
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
Scalar advection equation
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 advection
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)
Coupled advection equations
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
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
Admissibility / entropy conditions
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
Advection
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