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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.