Geodesic
Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 7, pp. 1282-1293

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An approach to the construction of second-and higher order accurate difference schemes in time and space is described for solving the linear one-and multidimensional advection equations with constant coefficients by the Godunov method with antidiffusion. The differential approximations for schemes of up to the fifth order are constructed and written. For multidimensional advection equations with constant coefficients, it is shown that Godunov schemes with splitting over spatial variables are preferable, since they have a smaller truncation error than schemes without splitting. The high resolution and efficiency of the difference schemes are demonstrated using test computations. 
@article{ZVMMF_2008_48_7_a11,
     author = {N. Ya. Moiseev and I. Yu. Silant'eva},
     title = {Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the {Godunov} method with antidiffusion},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1282--1293},
     publisher = {mathdoc},
     volume = {48},
     number = {7},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_7_a11/}
}
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N. Ya. Moiseev; I. Yu. Silant'eva. Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 7, pp. 1282-1293. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_7_a11/