doi: 10.17586/2226-1494-2015-15-4-731-740

# ONE-DIMENSIONAL GAS DYNAMICS PROBLEMS AND THEIR SOLUTION BASED ON HIGH-RESOLUTION FINITE DIFFERENCE SCHEMES

P. V. Bulat, K. N. Volkov

Article in Russian

For citation: Bulat P.V., Volkov K.N. Оne-dimensional gas dynamics problems and their solution based on high-resolution finite difference schemes. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol.15, no. 4, pp. 731–740.

Abstract
One-dimensional unsteady gas dynamics problems are revealing tests for the accuracy estimation of numerical solution with respect to simulation of supersonic flows of inviscid compressible gas. Numerical solution of Euler equations describing flows of inviscid compressible gas and conceding continuous and discontinuous solutions is considered. Discretization of Euler equations is based on finite volume method and WENO finite difference schemes. The numerical solutions computed are compared with the exact solution of Riemann problem. Monotonic correction of derivatives makes possible avoiding new extremes and ensures monotonicity of the numerical solution near the discontinuity, but it leads to the smoothness of the existing minimums and maximums and to the accuracy loss. Calculations with the use of WENO schemes give the possibility for obtaining accurate and monotonic solution with the presence of weak and strong gas dynamical discontinuities.

Keywords: gas dynamics, finite difference scheme, shock wave, rarefaction wave, contact discontinuity, Riemann problem, Sod problem, Lax problem.

References
1. Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu. Matematicheskie Voprosy Chislennogo Resheniya Giperbolicheskikh Sistem Uravnenii [Mathematical Problems in the Numerical Solution of Hyperbolic Systems]. Moscow, Fizmatlit Publ., 2001, 608 p.
2. Wesseling P. Principles of Computational Fluid Dynamics. Springer, 2000, 664 p. doi: 10.1007/978-3-642- 05146-3
3. Volkov K.N. Raznostnye skhemy rascheta potokov povyshennoi razreshayushchei sposobnosti i ikh primenenie dlya resheniya zadach gazovoi dinamiki [High-resolution difference schemes of flux calculation and their application to solving gas dynamics problems]. Vychislitel'nye Metody i Programmirovanie, 2005, vol. 6, no. 1, pp. 146–167.
4. Volkov K.N., Deryugin Yu.N., Emel'yanov V.N., Kozelkov A.S., Teterina I.V. Raznostnye Skhemy v Zadachakh Gazovoi Dinamiki na Nestrukturirovannykh Setkakh [Difference Schemes in Gas Dynamic Problems on Unstructured Grids]. Moscow, Fizmatlit Publ., 2014, 412 p.
5. Wolf W.R., Azevedo J.L.F. High-order ENO and WENO schemes for unstructured grids. International Journal for Numerical Methods in Fluids, 2007, vol. 55, no. 10, pp. 917–943. doi: 10.1002/fld.1469
6. Castro M., Costa B., Don W.S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. Journal of Computational Physics, 2011, vol. 230, no. 5, pp. 1766–1792. doi: 10.1016/j.jcp.2010.11.028
7. Clain S., Diot S., Loubere R. A high-order finite volume method for systems of conservation laws-multidimensional optimal order detection (MOOD). Journal of Computational Physics, 2011, vol. 230, no. 10, pp. 4028–4050. doi: 10.1016/j.jcp.2011.02.026
8. Hu G.H., Li R., Tang T. A robust WENO type finite volume solver for steady Euler equations on unstructured grids. Communications in Computational Physics, 2011, vol. 9, no. 3, pp. 627–648. doi: 10.4208/cicp.031109.080410s
9. Su X., Sasaki D., Kazuhiro N. Efficient implementation of WENO scheme on structured meshes. Proc. 25th
Computational Fluid Dynamics Symposium. Osaka, Japan, 2011, no. C01-3, 9 p.
10. Tsoutsanis P., Titarev V.A., Drikakis D. WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions. Journal of Computational Physics, 2011, vol. 230, no. 4, pp. 1585–1601. doi: 10.1016/j.jcp.2010.11.023
11. Vincent P.E., Castonguay P., Jameson A. A new class of high-order energy stable flux reconstruction schemes. Journal of Scientific Computing, 2011, vol. 47, no. 1, pp. 50–72. doi: 10.1007/s10915-010-9420-z
12. Rozhdestvenskii B.L., Yanenko N.N. Sistemy Kvazilineinykh Uravnenii i ikh Prilozheniya k Gazovoi Dinamike [Systems of Quasilinear Equations and their Applications to Gas Dynamics]. Moscow, Nauka Publ., 1978, 688 p.
13. van der Heul D.R., Vuik C., Wesseling P. A conservative pressure-correction method for flow at all speeds. Computers and Fluids, 2003, vol. 32, no. 8, pp. 1113–1132. doi: 10.1016/S0045-7930(02)00086-5
14. Xiao F. Unified formulation for compressible and incompressible flows by using multi-integrated moments. I. One-dimensional inviscid compressible flow. Journal of Computational Physics, 2004, vol. 195, no. 2, pp. 629–654. doi: 10.1016/j.jcp.2003.10.014
15. Sod G.A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation
laws. Journal of Computational Physics, 1978, vol. 27, no. 1, pp. 1–31. doi: 10.1016/0021-9991(78)90023-2
16. Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics, 1954, vol. 7, no. 1, pp. 159–193.
17. Arora M., Roe P.L. A well-behaved TVD limiter for high-resolution calculations of unsteady flow. Journal of Computational Physics, 1997, vol. 132, no. 1, pp. 3–11. doi: 10.1006/jcph.1996.5514
18. Woodward P.R., Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 1984, vol. 54, no. 1, pp. 115–173. doi: 10.1016/0021-9991(84)90142-6
19. Einfeldt B., Munz C.D., Roe P.L., Sjogren B. On Godunov-type methods near low densities. Journal of Computational Physics, 1991, vol. 92, no. 2, pp. 273–295. doi: 10.1016/0021-9991(91)90211-3
20. Sjogreen B., Yee H.C. Variable high order multiblock overlapping grid methods for mixed steady and unsteady multiscale viscous flow. Communications in Computational Physics, 2009, vol. 5, no. 2–4, pp. 730–744.