Group-Invariant Solutions of Hydrodynamics

The equations of hydrodynamics, being nonlinear, are in general difficult to solve analytically. A great deal of effort has therefore gone into the numerical solution of these equations, using a wide variety of algorithms. Issues associated with these numerical solutions include accuracy and stability of the algorithms and their associated solutions. Comparison to experiment is one basic way to address the validity of numerical solutions, but the issues of diagnostics and experimental error are always present. Further, there are regimes for which experimental results are either costly or impossible to obtain. Due to this, analytic solutions to such equations in relevant physical regimes have been sought. Such analytic, exact solutions can be used for three purposes: (1) benchmarks for numerical algorithms, (2) the basis for analytic models, (3) to provide insight into more general solutions.