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 
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/
@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},
     year = {2002},
     volume = {14},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2002_14_3_a4/}
}
TY  - JOUR
AU  - D. N. Bokov
TI  - Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function
JO  - Matematičeskoe modelirovanie
PY  - 2002
SP  - 43
EP  - 58
VL  - 14
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MM_2002_14_3_a4/
LA  - ru
ID  - MM_2002_14_3_a4
ER  - 
%0 Journal Article
%A D. N. Bokov
%T Characteristic directions technique of solving scalar one-dimensional nonlinear advection equation with noncovex flow function
%J Matematičeskoe modelirovanie
%D 2002
%P 43-58
%V 14
%N 3
%U http://geodesic.mathdoc.fr/item/MM_2002_14_3_a4/
%G ru
%F MM_2002_14_3_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, GITTL, M., 1950

[2] Oleinik O. A., “O zadache Koshi dlya nelineinykh uravnenii v klasse razryvnykh funktsii”, DAN, XCV:3 (1954), 451–454 | MR

[3] Godunov S. K., Vospominaniya o raznostnykh skhemakh, Nauchnaya kniga, Novosibirsk, 1997

[4] Gelfand I. M., “Nekotorye zadachi teorii kvazilineinykh uravnenii”, UMN, 14:2(86) (1959), 87–158 | MR

[5] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR | Zbl

[6] Oleinik O. A., “O edinstvennosti i ustoichivosti obobschennogo resheniya zadachi Koshi dlya kvazilineinogo uravneniya”, UMN, 14:2(86) (1959), 165–170 | MR

[7] Lindquist W. B., “The scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown”, Siam J. Math. Anal., 17:5 (1986), 1178–1197 | DOI | MR | Zbl

[8] Rozhdestvenskii B. L., “Razryvnye resheniya sistem kvazilineinykh uravnenii giperbolicheskogo tipa”, UMN, 15:6(96) (1960), 59–117 | MR | Zbl

[9] Samarskii A. A., “Raznostnaya approksimatsiya konvektivnogo perenosa s prostranstvennym rasschepleniem vremennoi proizvodnoi”, Matem. modelirovanie, 10:1 (1998), 86–100 | MR

[10] Kurganov A., Tadmor E., “New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations”, J. Comput. Phys., 160 (2000), 241–282 | DOI | MR | Zbl

[11] Shu C.-W., Osher S., “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, J. Comput. Phys., 77:2 (1988), 439–471 | DOI | MR | Zbl

[12] Harten A., Engquist B., Osher S., Chakravarthy S. R., “Uniformly high order accurate essentially nonoscillatory schemes, III”, J. Comput. Phys., 71:2 (1987), 231–303 | DOI | MR | Zbl