On the accuracy of shock-capturing schemes calculating gas-dynamic shock waves

A comparative experimental accuracy study of three shock-capturing schemes (the second-order CABARET, third-order Rusanov, and fifth-order in space third-order in time A-WENO schemes) is carried out by numerically solving a Cauchy problem with smooth periodic initial data for the Euler equations of gas dynamics. In the studied example, the solution breaks down and shock waves emerge. It is shown that the CABARET and A-WENO schemes, which are constructed using nonlinear limiters as a stabilization mechanism, have approximately the same accuracy in the areas of shock wave influence, while the nonmonotone Rusanov scheme has significantly higher accuracy in these areas despite producing noticeable nonphysical oscillations in the immediate vicinities of shock waves. At the same time, the combined scheme obtained based on the Rusanov and CABARET schemes localizes shock wave fronts, which are captured in a non-oscillatory manner, and preserves higher accuracy in the areas of the shock influence.