Geodesic
Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function
Matematičeskoe modelirovanie, Tome 14 (2002) no. 3, pp. 43-58

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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. 
@article{MM_2002_14_3_a4,
     author = {D. N. Bokov},
     title = {Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {43--58},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2002_14_3_a4/}
}
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D. N. Bokov. Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function. Matematičeskoe modelirovanie, Tome 14 (2002) no. 3, pp. 43-58. http://geodesic.mathdoc.fr/item/MM_2002_14_3_a4/