Hybrid running schemes with upwind and bicompact symmetric differencing for hyperbolic equations

New hybrid difference schemes are proposed for computing discontinuous solutions of hyperbolic equations. Involved in these schemes, a bicompact scheme that is third-order accurate in time and fourth-order accurate in space is monotonized using several partner schemes, namely, a first-order accurate explicit upwind scheme and two bicompact schemes of second and fourth orders of accuracy in space, both of the first order of accuracy in time. Their total domain of monotonicity covers all Courant numbers. An algorithm for automatically choosing the most suitable partner scheme is constructed. The mechanism of switching between high- and low-order accurate schemes is rigorously substantiated. All the schemes used can be efficiently implemented by applying the running calculation method. The hybrid schemes proposed have been tested on a model two-dimensional explosion problem in an ideal gas.