An adaptive multidomain Chebyshev method for nonlinear eigenvalue problems: Application to self-similar solutions of gas dynamics equations with nonlinear heat conduction

An adaptive multidomain Chebyshev collocation method for handling nonlinear eigenvalue problems, such as those arising in the computation of self-similar solutions, is presented. The algorithm is made of two stages, the second one using an iterative method. The method is applied to the time-dependent one-dimensional self-similar solutions of the Euler equations with nonlinear heat conduction in which a thermal front follows a shock wave. It leads to (highly) accurate solutions, for solutions stiffness ranging from 1 to 10−7. The method is quite general and may be applied to a large class of problems.