Geodesic
Smooth volume integral conservation law and method for problems in Lagrangian coordinates
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 473-484

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An integral conservation law is derived for smooth volume in Lagrangian coordinates (a comoving frame). A method for approximation of the integral smooth volume conservation law is discussed. An extension technique is suggested for development of smooth volume schemes. For hyperbolic systems, smooth volume upwind and Godunov schemes with monotonic reconstruction are derived. The schemes are applied to equations of gas dynamics and tested for three gas-dynamics shock tube problems. The solutions are monotonic and precise. 
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     author = {T. Ismagilov},
     title = {Smooth volume integral conservation law and method for problems in {Lagrangian} coordinates},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {473--484},
     publisher = {mathdoc},
     volume = {46},
     number = {3},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a10/}
}
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T. Ismagilov. Smooth volume integral conservation law and method for problems in Lagrangian coordinates. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 473-484. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a10/