On the accuracy of discontinuous Galerkin method calculating gas-dynamic shock waves

The results of a numerical calculation of gas-dynamic shock waves that arise when solving the Cauchy problem with smooth periodic initial data are presented using three variants of the DG (Discontinuous Galerkin) method, in which the solution is sought in the form of a piecewise linear discontinuous function. It is shown that the methods DG1A1 and DG1A2, for which the Cockburn limiter with parameters A1 = 1 and A2 = 2 are used for monotonization, have approximately the same accuracy in the influence areas of shocks (arising as a result of gradient catastrophes within the computational domain), while the nonmonotonic DG1 method, in which this limiter is not used, has a significantly higher accuracy in these areas, despite noticeable non-physical oscillations on shocks. With this in mind, the combined scheme obtained by the joint application of the DG1 and DG1A1 methods monotonously localizes the shocks and maintains increased accuracy in the areas of their influence.