On the accuracy of the discontinuous Galerkin method inside centered rarefaction waves and in the areas of their influence

The accuracy of two schemes of the discontinuous Galerkin method with piecewise linear solutions is investigated when calculating the special Cauchy problem for shallow water equations with discontinuous initial data, the exact solution of which contains a centered rarefaction wave and does not contain a shock wave. It is shown that inside the centered rarefaction wave and in the area of its influence, the solutions of both schemes with different orders converge to different invariants of the exact solution, which leads to a decrease in the accuracy of these schemes when calculating the vector of the basic variables of the considered Cauchy problem.