4.1.1 A coupled non-linear system

• The 1D hydrodynamics equations (40) are a non-linear system of 3 coupled equations.
• There is a little bit from the linear advection equation:
  • Derivation of the characteristic form (48).
  • Linear advection as special case (see Sect. 2.1.3).
• There is a little bit from Burgers' equation:
  • Shocks can arise from smooth initial conditions.
  • Discontinuities in the initial conditions can produce rarefaction waves.
  • We need to look for weak solutions (that satisfy an entropy condition).
• Characteristics are not necessarily straight lines.
• The domain of dependence is not only a single point but an interval. However, it is bounded. Therefore explicit methods (with limited full numerical domain of dependence) are often adequate.