.. _lectures: 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 :ref:`class_repos`, or in the Github repository `clawpack/fvmhp_materials `__, which also includes the latex files that created the slides. Lecture contents ^^^^^^^^^^^^^^^^ "FVMHP" refers to the :ref:`info_text`. 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 `Slides `__ ... `YouTube 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) `Slides `__ ... `YouTube video `__ FVMHP03 - Linearization of Nonlinear Systems --------------------------------------------- Material from FVMHP Chap. 2 - General form, Jacobian matrix - Scalar Burgers' equation - Compressible gas dynamics - Linear acoustics equations `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube video `__ FVMHP05 - Linear Systems - Riemann Problems ----------------------------------------------------- Material from FVMHP Chap. 3 - Riemann problems - Riemann problem for advection - Riemann problem for acoustics - Phase plane `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube video `__ FVMHP13 - Nonlinear scalar rarefaction waves ----------------------------------------------------- Material from FVMHP Chap. 11 - Form of centered rarefaction wave - Non-uniqueness of weak solutions - Entropy conditions `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube video `__ FVMHP22 - Multidimensional hyperbolic problems ----------------------------------------------- Material from FVMHP Chap. 18 - Derivation of conservation law - Hyperbolicity - Advection - Gas dynamics and acoustics - Shear waves `Slides `__ ... `YouTube 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 $BAq_{xy}$ `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube 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 `Slides `__ ... `YouTube video `__