Geodesic
The third-order approximation Godunov method for the equations of gas dynamics
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 1, pp. 71-83.

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The paper suggests a new modification of Godunov difference method with the 3rd order approximation in space and time for hyperbolic systems of conservation laws. The difference scheme uses the simultaneous discretization of the equations in space and time without of Runge — Kutta stages. An exact or approximate solution of Riemann problem is applied to calculate numerical fluxes between cells. Before the time step, corrections to the arguments of the Riemann problem providing third-order approximations for linear systems are calculated. After the time step, the numerical solution correction procedure is applied to eliminate the second-order error arising from the nonlinearity of the equations. The paper presents the results of experimental numerical verification of the method approximation order on the exact smooth solution inside the fan of the expansion wave. The test results completely confirm the third order of the presented method. The proposed approach of constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.
Keywords: nonlinear hyperbolic systems, Godunov method, third order, approximation, construction of difference schemes. 
@article{VVGUM_2019_22_1_a5,
     author = {E. I. Vasilev and T. A. Vasilyeva and D. I. Kolybelkin and B. G. Krasovitov},
     title = {The third-order approximation {Godunov} method for the equations of gas dynamics},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {71--83},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2019_22_1_a5/}
}
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E. I. Vasilev; T. A. Vasilyeva; D. I. Kolybelkin; B. G. Krasovitov. The third-order approximation Godunov method for the equations of gas dynamics. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 1, pp. 71-83. http://geodesic.mathdoc.fr/item/VVGUM_2019_22_1_a5/