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In this section, we'd like to analyze nonlinear hyperbolic equations.
The simplest problem is a scalar one that can be stated as:
As in the linear case, information is transported along the characteristic
lines. However, this is not the most important feature to analyze in here.
When solving these problems it is usual to find discontinuous solutions, even
with smooth initial conditions. Therefore, it is necessary to develop
numerical strategies that permit finding solutions for problems with shocks.
A classical example of nonlinear hyperbolic problems is
Burgers' equation,
obtained using  .
Even for this simple case, solution becomes discontinuous if a decreasing initial data is considered.
The problem is harder when trying to solve Euler equations, which are equations
governing a compressible flow problem. These equations form a nonlinear hyperbolic
system that can be stated as:
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Unsteady Transport