Mathematics / advanced
Hyperbolic equations and propagation
Track finite-speed propagation, characteristic structure, energy, discontinuities, and physically selected weak solutions.
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Inside this module
Module 01 advanced
Hyperbolic equations and propagation
Track finite-speed propagation, characteristic structure, energy, discontinuities, and physically selected weak solutions.
- 01
Characteristic surfaces and hyperbolicity
Read propagation directions from principal symbols while distinguishing scalar characteristic roots from a system's well-posedness structure.
33 min - 02
Wave-equation energy and domain of dependence
Use a local energy balance on shrinking balls to prove finite propagation for the constant-speed wave equation in any spatial dimension.
34 min - 03
d'Alembert, Kirchhoff, and fundamental solutions
Compare explicit free-wave formulas in one and three dimensions and explain where dimension-dependent tails enter.
35 min - 04
First-order symmetric hyperbolic systems
Construct a positive energy from the system matrices and diagonalize a linear acoustic example into counterpropagating fields.
34 min - 05
Scalar conservation laws and characteristics
Follow state-dependent characteristic speeds until the initial-label map loses invertibility and gradients become singular.
34 min - 06
Shock formation and Rankine–Hugoniot
Continue conservation across a moving jump using a signed distributional balance, without confusing that balance with entropy admissibility.
34 min - 07
Entropy conditions and Kružkov solutions
Select bounded scalar weak solutions with a full entropy family and state the assumptions behind uniqueness and L¹ contraction.
36 min - 08
Riemann problems and rarefaction waves
Solve a convex scalar jump problem by shock or a self-similar fan and evaluate the state selected at a fixed interface.
34 min - 09
Systems of conservation laws and wave families
Decompose coupled perturbations into characteristic families and state the local hypotheses behind nonlinear Riemann constructions.
36 min - 10
Finite-volume fluxes and discrete entropy
Separate conservation, flux consistency, monotonicity, and entropy dissipation in a fully discrete finite-volume update.
36 min