Titlebook: Numerical Methods for Conservation Laws; Randall J. LeVeque Book 19901st edition Birkhäuser Basel 1990 numerical method.research.shock wav

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Shocks and the Hugoniot LocusWe now return to the nonlinear system .. + .(.). = 0, where .(., .) ∈ ℝ..

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Conservative Methods for Nonlinear ProblemsWhen we attempt to solve nonlinear conservation laws numerically we run into additional difficulties not seen in the linear equation. Moreover, the nonlinearity makes everything harder to analyze. In spite of this, a great deal of progress has been made in recent years.

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Approximate Riemann SolversGodunov’s method, and higher order variations of the method to be discussed later, require the solution of Riemann problems at every cell boundary in each time step. Although in theory these Riemann problems can be solved, in practice doing so is expensive, and typically requires some iteration for nonlinear equations.

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