28. Hyperbolic equations#

  • Integral equations

  • Jump conditions

  • Differential equations

  • Method of characteristics: domains of influence

  • “Inviscid” limit, multiple solutions, entropy condition and unique physical solution

Some useful script on GDrive \(\texttt{basics-books/colabs/math/pde/fvm}\). So far 1-dimensional problem only: linear equation, linear system (linearized p-sys), non linear system (p-sys).

Problems

1-dimensional domain

n-dimensional domain

  • Scalar linear

  • Vector linear

  • Scalar non-linear

  • Vector non-linear

todo

  • hyperbolic problems as the ideal limit of dissipative (parabolic?) problems. In non-linear problems, multiple (infinite?) solutions of the hyperbolic problem exist, but there’s only one (? Prove it! Or just provide the idea) solution that is the limit of the solution of the dissipative problem, in the ideal limit: when do jumps occur? When do rerefaction fans occur?

    • show multiple solutions

    • show limit of the dissipative solution

    on some simple problems, maybe starting from a scalar problem in 1-dimesnional domain, and then generalize to “more difficult” problems

  • domains of dependence and influence. Show them in both unsteady and steady problems?

    • Example with compressible flows (or the p-sys) comparing unsteady and steady problems, and discussing the nature of the equations in the subsonic and supersonic regimes