Analysis of different modiҥcations of the CABARET scheme applied for approximation of the system of gas-dynamic equations

Abstract

На основе метода квазиинвариантов построены различные модификации схемы CABARET, аппроксимирующие систему уравнений газовой динамики неизоэнтропических течений. Проведен сравнительный анализ точности этих модификаций при расчете трех различных задач Римана (в частности, задач Сода и Лакса) о распаде начального разрыва в политропном газе. Показано, что некоторые модификации схемы CABARET могут иметь существенные преимущества по сравнению с другими модификациями этой схемы относительно различных критериев, связанных с оптимальной локализацией сильных и слабых разрывов точного решения. In this paper, we construct various modifications of the CABARET scheme that approximate the system of nonisentropic gas dynamics equations. The construction of these modifications relies on representation of conservation laws using various non-divergent forms and using the method of quasi-invariants. Each family of these invariants corresponds to its own modiҥcation of the CABARET scheme. The algorithm of the resulting CABARET scheme is given for the case when in the calculated exact solution the characteristics of one family are unidirectional, i. e. velocity distribution of these characteristics does not change sign. A comparative analysis for the accuracy of these modiҥcations was carried out when calculating the Sod problem on the attenuation of an initial discontinuity in a polytropic gas. Based on this analysis, a modification was chosen that most accurately localizes strong and weak discontinuities in the exact solution and has the smallest integral imbalance and the best order of convergence. Further, this modification of the CABARET scheme was compared with the best modification obtained from the previous work on various test problems (including the Lax problem) without changing the sign of the rate characteristics. The given complex analysis has shown that the new modification has better convergence compared to the modification from the previous work. Also, using the example of the Riemann problem of two divergent strong discontinuities, it was shown that it is impossible to correctly determine the order of local convergence of the numerical solution to the exact one in zones of constant ѕows located in the areas of inѕuence of shock waves in the considered modifications of the CABARET scheme.