A comparison of two approaches to construct combined schemes for multidimensional Euler equations

A substantial disadvantage of standard numerical schemes with nonlinear flux correction is a severe deterioration in accuracy in influence areas of shock waves: a drop in the order of convergence to the first and a significant increase in solution errors. Combined numerical schemes make it possible to solve this problem and at the same time maintain the monotonicity of the calculated solution. The main element of any combined scheme is a nonmonotone scheme, which has high accuracy in shock waves influence areas. Two approaches to constructing such schemes are compared. The first is a local increase in the order of approximation in time using Runge-Kutta or Adams methods. The second is a global, a posteriori increase in the order of accuracy in time using Richardson methods. The comparison is carried out for schemes with a bicompact spatial approximation of the fourth order, using a test case for two-dimensional Euler equations. Its solution is a periodic smooth isentropic wave, which over time undergoes a gradient catastrophe and turns into a periodic shock wave. It is shown that the first approach is preferable if the problem allows to be solved by a nonmonotone scheme completely, and meshes are moderately refined.