Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function

A concept of characteristic directions technique to solving nonlinear advection equation is presented. Two meshes: characteristic and Eulerian are used. A characteristic mesh is adaptive both to the properties of the initial distribution function and to the properties of the boundary condition function. This allows: development of the algorithm for obtaining a numerical solution on characteristic mesh using the properties of the solution of nonlinear advection equation in smooth region; to determine the configuration and the solution at the arbitrary discontinuity decay; to reproduce spatial location and solution value at the discontinuity points and extreme points at the accuracy determined by interpolation and approximation of initial values and boundary condition functions.