van Leer, Bram Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. (English) Zbl 1364.65223 J. Comput. Phys. 32, 101-136 (1979). Summary: A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov’s method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov’s method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the ultimate conservative difference scheme” (for Parts I–IV, see [Zbl 0255.76064; Zbl 0276.65055; Zbl 0339.76039; Zbl 0339.76056]). Cited in 4 ReviewsCited in 1381 Documents MSC: 65N06 Finite difference methods for boundary value problems involving PDEs 76M20 Finite difference methods applied to problems in fluid mechanics Citations:Zbl 0255.76064; Zbl 0276.65055; Zbl 0339.76039; Zbl 0339.76056 PDFBibTeX XMLCite \textit{B. van Leer}, J. Comput. Phys. 32, 101--136 (1979; Zbl 1364.65223) Full Text: DOI References: [1] Godunov, S. K., Mat. Sb., 47, 271 (1959), also: Cornell Aeronautical Lab. Transl. [2] van Leer, B., J. Computational Physics, 23, 276 (1977) [3] Boris, J. P., U.S. Naval Research Lab. Memorandum Report 2542 (December 1972) [4] Godunov, S. K.; Zabrodyn, A. W.; Prokopov, G. P., Z. Vyčisl. Mat. i Mat. Fiz., 1, 1020 (1961), also: Cornell Aeronautical Lab. Transl. [5] van Leer, B., J. Computational Physics, 23, 263 (1977) [6] Strang, W. G., SIAM J. Numer. Anal., 5, 506 (1968) [7] Noh, W. F.; Woodward, P. R., (Lecture Notes in Physics No. 59 (1977), Springer-Verlag: Springer-Verlag Berlin), 330 [8] P. R. Woodward; P. R. Woodward [9] Weber, W. J.; Boris, J. P.; Gardner, J., ALFVEN, a two-dimensional SHASTA code solving the radiative, diffusive MHD equations, Comput. Phys. Comm., 16, 243 ] (1979) [10] Boris, J. P., U. S. Naval Research Lab. Memorandum Report 3237 (March 1976) [11] Fromm, J. E., J. Computational Physics, 3, 176 (1968) [12] van Leer, B., (Lecture Notes in Physics No. 18 (1973), Springer-Verlag: Springer-Verlag Berlin), 163 [13] van Leer, B., J. Computational Physics, 14, 361 (1974) [14] Courant, R.; Isaacson, E.; Rees, M., Comm. Pure Appl. Math., 5, 243 (1952) [15] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience: Interscience New York · Zbl 0041.11302 [16] P. R. Woodward; P. R. Woodward [17] van Leer, B., The characteristic equations for discontinuous flow, appendix to original version of present paper (October 1977) [18] Sod, G. A., J. Computational Physics, 27, 1 (1978) [19] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502 [20] Fromm, J. E., IBM Research Report RJ 780 (1970) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.